I never leave my horse's halters on for turnout or while they are in their stalls. I don't want to take the chance that they could get caught on something, even if it is a very small chance. I also wouldn't want them to get rubs on their faces from having to wear them all of the time.
If you need to leave a halter on your horse, for whatever reason that may be, it really should be a leather halter or a nylon halter with a leather crown piece and it should be fitted to your horse properly. The leather halter would be the better choice of the two. The leather should break if your horse gets caught on something. In my opinion it would be a very bad idea to have a horse turned out or stalled in an all nylon halter (even though I know people who do) and an even worse idea to turn out in a rope halter.