Leather only, always, for safety reasons. I don't care if it breaks as long as the horse doesn't, the halter can be replaced. And I'm a snob, I don't wear polyester and don't expect the horse to wear nylon.
I have seen so many horses get hurt when they pull back, and a nylon halter doesn't release . However - my horse, who likes to break halters for fun...enjoys ESPECIALLY to break the nice *expensive* ones I buy, not the cheapy ones.
My horse has a nylon halter with leather crown and a spare (old, has broken easily in the past but is still usable until he breaks the crownpiece again and it won't be long enough for me to punch another hole in) that's all leather. He's not turned out with one on. I don't like to turn out in all nylon halters, although I've never had a bad experience with them. It's just one less thing to worry about.
Nylon with breakaway strap for home (leading, tying, etc) and leather for shows. I used a leather halter for home, too, until my horse figured out how to take his halter off of the gate during turnout and chewed through a succession of leather halters. That is why we can't have nice things.
"I'm not always sarcastic. Sometimes I'm asleep."
- Harry Dresden