To make it easier for reference later, please post one location/trail head at a time with the name at the top. If there are things people should know about the area, please be sure to include that as well (rules, conditions, etc.).
Probably the Linear Park/Stub Stewart park trails. We usually start out at the Buxton trail head and head towards Stub Stewart. If you go in the other direction the trail only goes for about two miles before it hits a shady looking gravel road. There's one house along the way that has some sign about shooting trespassers or something. I turned around after I saw that!
If you follow the trail a few miles in the good direction you'll eventually cross the road that leads into Stub Stewart. I keep Heffner leashed when we're at the trailhead and crossing any roads because it's technically an on-leash area. I've never seen another dog stay on leash there though. The only thing to watch out for are horses if your dog isn't a fan. Horses are allowed on the trail as well.
best every day hike - hands down - the Wildwood, starting at the trailhead on NW Germantown Road
it's about 10 minutes away, and has miles and miles and miles of options
it can be very muddy in the rainy season, though
it is leash-required, as are most places, but most runners/hikers I see on the trail have dogs, and most of those dogs are off-leash. caveat being that most of the dogs are more well-behaved than is the norm.
while some of the trails allow bikes/horses, the wildwood itself is foot traffic only.