United gave me a really hard time when I flew with a mountain dulcimer several years ago. They didn't like the soft case because it "wouldn't protect the instrument in the overhead compartment" (it was too long to go under the seat), but if I put it in the wooden box, it was too big and they made me check it. I wrapped it in several layers of cardboard and wrote "FRAGILE" all over it but they still managed to smash the end of the box. The dulcimer was fine, but if it had been damaged, their insurance only covered losses (as in, disappearances) and I would have been SOL.
I've never tried to fly with a guitar--the Guild has a hard case but it's old and I don't want to risk it getting chewed up if they make me check it, and the Alvarez only has a padded nylon deal, which would probably be OK in the overhead as long as people weren't overzealous about cramming in other stuff.
I don't fly much, but what kinds of cases seem to be best accepted? Frankly, my brother had less trouble getting in with a full-sized gun case than I did with dulcimers.