I have favorites, but cmon, in the WORLD? How much do we really know about what's out there? I probably wouldn't pick a popular one. If I were going to pick one, maybe the Pacific Crest, because it goes through all kinds of country. So does the AZT. I probably would not pick a trail where a permit is required.
As for a short trail, I have my memories of Fish Creek Trail #60, 12 miles long, from just south of Hannagan Meadow Lodge on down to the Black River. I used to tell backpacker friends about this trail over the years, and urged them to skip Bear Wallow, forget about Bear Wallow and go to Fish Creek #60 instead. And most of them ignored me and went to heavily beaten Bear Wallow because it's very well-known, it's in a designated wilderness (although only a couple of miles from Fish Creek, which isn't in a designated wilderness.) You know, it's so important to hike trails because they are famous.
But it's too late now. That area was just about the most heavily burned in the Wallow Fire in 2011. Unlike most other trails which have reopened since the fire, that one has not, and it will remain closed for many years, I've been told.
I'm glad I went there, though. It was my favorite trail. You could have set up camp down there all summer and you would have seen just a few people. Plus, again much better than Bear Wallow, it went to the Black River without having to cross onto the Apache reservation. When you got to the river you could turn upstream and be at a nice deep swimming hole in half a mile or so. You had a good chance while there to see elk, deer, bears, mountain lions, wolves, Rocky Mountain bighorns, and a host of smaller fauna. And a good chance of not seeing people. The trail was there, but it was kind of faint due to lack of traffic...
http://hikearizona.com/decoder.php?ZTN=327
http://hikearizona.com/photo.php?ZIP=43714