At virtually every road crossing you see a sign that says foot traffic only. Sometimes they list no horses, no motorized vehicles etc.
Today, I met a local guy who saw horses on the AT a few weeks ago. A couple was driving home with their horse trailer and had car trouble. Turns out their property is very near the AT and only a few miles from where they broke down. So they called the Forest Service and got permission to ride their horses on the AT as it was the safest way to get the horses home. I thought it was a cool story.