After this question was asked in the dailies, I thought it would be a good thread to keep around to help us as we travel around the country.  This will help answer the questions of "where do I run when I go to (blank)?" .  Thanks, Mandy for pinning this.

Please add your trail system (with a link) and I'll add them to the alphabetized list. 

Multi-State

Appalachian Trail - great website for parking information for locating trail heads.

Pacific Crest Trail

Alaska

Our local non-motorized, multi-use trails - semi-urban spaghetti loop trails, used by almost anything without a motor - run, dog walk, mtn bike, equestrian, geocache, orienteer. XC ski and snowshoe in winter.  I point you to the general map page, but there's a bit more detail on the emergency planning maps under the Events page. 24-hr run race is held here.

Matanuska Greenbelt maps

Home of the Resurrection Pass 50/100 mi races.

Resurrection Pass trails

An assortment of trails are here in the Hatcher Pass Management Area - perhaps some of the best scenery around. Not sure that there's any official trail maps, although there may be some in the works. Hatcher Pass marathon is held on west-side road.

Arizona 

The Kachina Wilderness north of Flagstaff is a gorgeous part of the state.  As it is a 2+ hour drive for me and it is snow-packed 6 months per year, I only get to run it a couple of times per year.  It is definitely one of my favorites.  Most people choose the Humphrey's Peak Trailhead at the AZ Sno Bowl but I prefer the Inner Basin route.  At 16 miles round trip it is 6 miles longer but you'll see far less people until the 2 trails meet up for the last mile to the summit.

Phoenix is surrounded by mountains in virtually every direction and the county park system has taken advantage of that with some amazing trails for the locals.  Additionally The Maricopa Trail (as of yet unfinished) links them altogether for a 242+ trails encircling all of Phoenix.  As it is only a 15 minute drive from the house, The White Tanks west of Phoenix is my personal favorite with 25 miles of trails.  It is has some challenging trails with good climbs and it is probably the least crowded of the county parks.  Bonus: it isn't uncommon to see mule deer once you get a couple of miles in.

California

Hirshman Trail...I work at the government center at the end of the trail and this has been my daily lunch run.

My morning runs are bootleg trails that take me into Empire Mine  I can put together a nice 16 mile loop running through Empire Mine which is nice because of the hills it's challenging so you get a work out like a 20 mile run and no driving.  Longer runs are done else where.

Highly recommend the Marin Headlands, with miles and miles of trails and hills that will take you from the Golden Gate Bridge up the coast to Stinson Beach and beyond.  Many races here (Miwok 100k, North Face Endurance Challenge, Dipsea, lots of smaller events by local outfits)

Monterey/Carmel Area.   Fort Ord National Monument is my favorite, though it is becoming more popular (crowded) since it became a national monument.   Garland Ranch Regional Park has two separate peaks, both with 2000' of elevation.   Toro Park is smaller, but has significant climbs and splendid views.     And most frequently.... I run from home through the Del Monte Forest in Pebble Beach.  Over 20 miles of trails with occasional fantastic ocean views and jaunts next to fancy golf courses.

Humboldt County, California -

Arcata Community Forest - Partially situated behind Humboldt State University.  There is nothing flat about The Forest.  Plans are in motion to expand it.  No opportunity to refill water supplies, so be sure to take what you need.

Headwaters Forest Reserve - 11.5 mile round-trip.  1st mile is paved/ADD compliant, next 2 miles are well-groomed, undulating trails.  Then there's a 2-mile climb to the top where you will find an incredible stand of old growth redwood.  A half-mile loop takes you through a portion of the stand, then you can have a blast flying back down the 2 miles, then 3 miles out.  No opportunity to refill water supplies, so be sure to take what you need.

Prairie Creek Redwoods State Park - One of my absolute favorite places.  Run the approximately 5-mile James Irvine Trail to stunning Fern Canyon.  Along the way, you may find yourself face-to-face with an Roosevelt Elk, or venture further out to the parking area and dirt road beyond and you may find a herd of elk lounging and relaxing.  There are approximately 5-6 trail options to connect and wind your way around.  No opportunity refill water supplies unless you go back to park headquarters, so plan appropriately.  NOTE - You can access the trails from the Fern Canyon side, but there is a $5 entrance fee.  If you go to the park headquarters at Prairie Creek, there's camping and no charge for day use.

Colorado

Pawnee - Buchanan Loop

Connecticut

If you go to Connecticut, a huge interactive map of trails throughout the state can be found here:

https://www.ctwoodlands.org/blue-blazed-hiking-trails/blue-blazed-hiking-trails-interactive-map

District of Columbia

Rock Creek Park-- lots of options and who would have thought such a slice of wilderness would be right in the middle of DC?  Unless it's a gorgeous weekend day,  you can usually have the trails (mostly) to yourself.  Same thing with my other go to trail system on the outside of the Beltway, Northwest Branch Park.

Kansas

Wyandotte County Lake Park - This park has a 10.35 mile bridle loop and probably enough single track trails to make a 15 mile loop. It is located in Kansas City, KS and is the location of two 50Ks and numerous shorter races put on by the Trail Nerds.

Clinton Lake State Park - This park is the location of the Hawk Hundred, Free State Trail Runs, and several other races. It is a beautiful place to run. It is more technical than the Wyandotte County Lake trails, but not quite as hilly.

Lake Perry Trails - These trails are home to two 50Ks and a soon to be 50 miler that are put on by the Trail Nerds. The trails are similar to the Clinton Lake trails, rocky, but not quite as hilly as the Wyandotte County Lake trails. It is another beautiful place to run.

Shawnee Mission Park - This park has several interconnecting trails which can be combined to get in about a 9 mile run. It's a beautiful place to run.

Louisiana

Clark Creek-- Vertical. I think the guy in the link below must have spent a good bit of time in the creek bottom if he only got 1641 in the 4.7 mile loop. I have a friend who has it gps'd at 2050. Either way, you won't get any more rugged vertical gain anywhere close to here.

Maryland 

Loch Raven Reservoir is where we run most often here.  We also run in Gunpowder State Park sometimes.

Catoctin Mountain Park - Nice views, part of the Catoctin 50k Course.

Seneca Creek Greenway Trail - used by the Stone Mill 50 and he Seneca Creek 50k

Mississippi

Clear Springs Mississippi-- Pretty good long runs with decent ups and downs, mostly rolling but some steep stuff.

Montana

Helena, MT lots to choose from- short/ long; steep/relatively flat , out and back/loop outside of town, too many to list- National Forest in every direction

Missoula, MT:  We are surrounded by trails.  You literally cannot head in any direction from Missoula and not run into some fantastic, single track, mountain trails.  There really are too many to list.  Here is a nice little guide to many of the local trails: http://www.missoulian.com/app/hikebike/  If it's your first trip to Missoula, I'd recommend hitting theRattlesnake area.

New Jersey

There are quite a few parks in NJ with runnable trails, and the best place I've found to locate them is this Biking website.

Ringwood State Park/Ramapo Valley Reservation: Park at the top of Skyline drive, or down in the Ramapo Valley Reservation  Lots of rocks throughout make for a challenging run no matter where you start. Site of Njtrail series Mountain Madness.. Bring a map.

South Mountain Reservation:  Accessible by train from NYC

Mercer Park:  Nice and flat dirt single track trails.

Merrill Creek Reservoir - This is my go to spot as it's only 5 minutes from home.  Mix of single track and fire roads.

New York

No super long trails close by, but this one's a favorite- pretty flat, but incredibly technical. The link is to the Local MTB Clubs info on it. They built it, they trash it, they fix it. I have mixed feelings:

There are some wonderful, pristine upland forest sections along the 500+ mile Finger Lakes Trail. There are also sketchy and road sections and a fair amount of hunting closures. I'll add to recommended sections as I have good gpx tracking for them, but here's the main site with all of the details.

Then there's a wonderful section I run every year when I'm down in Allegany State Park (best described as foothills to the Allegheny Mountains, and don't question the spelling!). Most of this is FLT, but it stays in the park as a point to point route. I will now digress, as while in the park you may want to take advantage of the most fun trail it has to offer, Beehunter. Best of all, no MTB allowed, Nyah nyah.

Back on the FLT, Conservation branch, an excellent section goes through Ellicottville, boasting both the occasionally stunning view and ridiculous hill climbing. I'm told that south of this there's another great section that slowly climbs through cushy pine forest, and I know that north of it there's some of the best singletrack I've ever run on in the McCarty Hill State Forest, but I don't have good data for it atm.

A bit further north of this is another section I run often, really nice, part road through here but I mention it because it crosses rte 20A close to my house. If you head north along the FLT from here, follow 20A east at the crossing up to the first drive past the diner and I'll show you around town.

North Carolina

Pigsah - Asheville, Henderson,  Brevard area

Route around Echo Mountain

Pinehurst / Southern Pines Area -  Weymouth Woods Sandhills Nature Preserve is a nice trail system with rolling hills and mix of rooted trails and sandy horse trails.  Running outside perimeter you can get 5 miles in a loop.

North Dakota

The Maah Daah Hey Trail. It's a 150 mile single-track mountain bike trail open to hikers, runners, walkers, horseback riders, and bikers. It's point-to-point through the North Dakota Badlands

Oregon

Rogue River Trail - 40 miles of remote point to point

I did this a few weeks ago with a friend and it was harder than R2R last year.

Pennsylvania

Michaux Forest - The maps section is great.  AT runs through here for almost 40 miles

Darlington Trail - - hard to get to in bad road conditions based upon where the trail heads are located

Boyd Big Tree Preserve  - good winter running

The "Buzzards" Course -   Not one I get to much but one of my favs.  I plan on getting out there for the baby buzzards (16 miles) in the next 3-4 weeks.

French Creek I use as a no brainer place for a weekend long run when I don't feel like travelling. I can get 3-5 hours in there easy without repeating any sections. I also have many 8-10 mile loops over there for evening runs in the lighter months. French Creek is also adjacent to Birdsboro Waters, so can hook up there for more miles and variety.

French Creek State Park

Coventry Woods is a go to place for shorter weekday runs with some fantastic climbs but also fast sections. I have some 6-7 mile loops there, plus some hilly twisty 8-12 mile out/back routes.

Coventry Woods Park

My go to trail is the one that the Dam Half marathon is run on. A few other trails in the area that I can add in to get a nice long run in. Only downside is it's difficult to totally avoid some roads.

Rothrock State Forest, starting at the Tussey Mountain Ski Area (off US 322 just east of Boalsburg), or at Shingletown (from PA 45 turn onto Mountain Road in Shingletown, follow it to the end, there's a small parking lot - daylight use only - and the trailhead), combining the trails there in wild varieties. Be prepared for steep, technical, rocky and rooty trails and some occasional mud puddles. Rattlesnakes very common on Bald Knob Mountain in warm months, be watchful.

Trails can be combined with gravel roads, which lead to different state parks (e.g. Whipple Dam, Alan Seeger, Greenwood Furnace, Penn Roosevelt,...), or just through the forest, for many miles.

Bellefonte Central Rail Trail is a mostly flat, non-technical trail that starts in State College, at Sunset Park (off Park Avenue, at the end of McKee Road - parking area is daylight use only). Run down the hill, turn right at the bottom, then follow the trail. Gravel surface, turning into grass later. 4.1 miles to Toftrees (where the trail ends), mile 2 is near/along highway, separated by wooded area. Ends at Toftrees - out and back is 8.25 miles.

Green Lane Reservoir is another go to place for shorter weekday runs (5-8 miles) or a longer run (13-16) around the entire lake. During hunting season it's the place to go as it's a no hunting park.

Green Lane Reservoir

Trexler Nature Preserve - a gem of a place surrounding the Lehigh Valley Zoo. A 8-10 mile loop on singletrack with multiple spur trails. Some gorgeous vistas a lot of incline and hills and you can run along and see the bison and elk sections of the zoo.

Valley Forge - Beautiful and historic place. Miles of bike paths if you like that type of running. But the singletrack is what shines here. Mt Joy and Mt Misery 4 mile loop is almost 1,000 vertical foot gain alone. The Chapel trail is also nice singletrack and that runs for a few miles from near Washington's Headquarters to the 422 bridge crossing the Schuylkill River to the Schuylkill River trail which runs to Philadelphia if you really want a long run. Also connects to the Horseshoe Trail that runs to Harrisburg and the AT. So possibilities are endless. 

Jacobsburg State Park - Nice mix of terrain.  Popular with horse back riders so you'll need to dodge some horse poop.

Lehigh Gap Nature Center - Former superfund site.  The AT runs through there as does the D & L Trail.

D & L Trail - Flat canal towpath and rail trails.  Stretches over 100 miles through Eastern PA.  If you're looking for an easy run without dealing with vehicular traffic this is a good option.

Wisconsin 

I run a lot of miles on various sections of the Ice Age National Scenic Trail.

International

South Korea

There is a trail system with a trailhead about 1/2 mile run from my apartmeapartnt. My go to run is the first 5 miles up the big hill of the run shown here.  This plot is my go to 17 miler. (Maybe you will need to switch to satilight mode to see it.) http://www.mapmyrun.com/routes/view/596049838

AT, I will sticky this if you don't mind being the keeper of it when people add stuff.  And if you will alphabetize the states...

In CA:

Highly recommend the Marin Headlands, with miles and miles of trails and hills that will take you from the Golden Gate Bridge up the coast to Stinson Beach and beyond.  Many races here (Miwok 100k, North Face Endurance Challenge, Dipsea, lots of smaller events by local outfits) http://www.parksconservancy.org/map/#/visit/park-sites/marin-headlands.html?coords=14:37.8308:-122.5244&date=1424922107312

Thanks, Gatsby and FSocks. The list is growing

Missoula, MT:  We are surrounded by trails.  You literally cannot head in any direction from Missoula and not run into some fantastic, single track, mountain trails.  There really are too many to list.  Here is a nice little guide to many of the local trails: http://www.missoulian.com/app/hikebike/  If it's your first trip to Missoula, I'd recommend hitting the Rattlesnake area.

There are quite a few parks in NJ with runnable trails, and the best place I've found to locate them is this Biking website.

My Favorites:

Ringwood State Park/Ramapo Valley Reservation: Park at the top of Skyline drive, or down in the Ramapo Valley Reservation  Lots of rocks throughout make for a challenging run no matter where you start. Site of Njtrail series Mountain Madness.. Bring a map.

South Mountain Reservation:  Accessible by train from NYC

Mercer Park:  Nice and flat dirt single track trails.

A couple from Alaska

Our local non-motorized, multi-use trails - semi-urban spaghetti loop trails, used by almost anything without a motor - run, dog walk, mtn bike, equestrian, geocache, orienteer. XC ski and snowshoe in winter.  I point you to the general map page, but there's a bit more detail on the emergency planning maps under the Events page. 24-hr run race is held here.

Matanuska Greenbelt maps

Home of the Resurrection Pass 50/100 mi races.

Resurrection Pass trails

An assortment of trails are here in the Hatcher Pass Management Area - perhaps some of the best scenery around. Not sure that there's any official trail maps, although there may be some in the works. Hatcher Pass marathon is held on west-side road.

Updated, thanks.