This article is from the Hiking/Backpacking With Canines, by Terri Watson with numerous contributions by others.
Unfortunately, uncontrolled dogs and irresponsible pet owners have
contributed to the closing of trails in a number of places. The
following are rules of thumb for picking a trail or area, but since
there are exceptions to must of these, the only way not to be
surprised is to call the correct admistrative office and check to be
certain that dogs are allowed. It would be quite dissappointing to
drive several hours to your selected trailhead only to find that dogs
are not allowed. The forest service offices are often a good place to
start, and they should be able to direct you to other numbers as
Dogs are not allowed on National Park or National Monument trails.
On-leash dogs are permitted on or near the paved, developed areas, but
that's all. National Forests often allow dogs on their trails, but
there are exceptions, so check first. Dogs are usually allowed on
wilderness area trails, but again, check to be sure.