More recently known as shelters, I prefer the old name fashioned after the classic Adirondack LT. The ATC has a page devoted to the shelters and suggests the positives and negatives as well as Shelter Etiquette!
High Point Number 1 LT, NJ. at 1230 miles from Springer Mtn.
For a list and map of the AT Shelters, look no further than here.
On my 2,000 mile I stayed in lean-tos probably 79 times. Not so much in Georgia, though, since I was in synched with the gaps and not the ridge lines.
Gentian Pond LT, NH at 1766 miles from Springer and brand new in 1974!
The AMC crew was leaving the site and left us a bunch of food.
Most LT were nice, (Locust Ridge LT),
some had resident mice, (Russell Field LT),
some leaked (Neys LT),
many were trashed by hikers (Curley Maple LT),
and a picnic table was always nice! (Old Orchard LT).
Now for two great references one free and one not. For example search for this title in the Whites.
Plus this one on a slide sharing website.
For a nice coffee table book, see The Appalachian Trail: Backcountry Shelters, Lean-Tos, and Huts