There is not a specific headstall you need for barrel racing. A headstall, is a headstall. There aren't "reining headstalls" or "barrel racing headstalls". Just a headstall.
Now myself personally, I want a browband headstall with a throatlatch, because I want that thing as secure as possible when I'm running breakneck speed through the pattern. I don't like one-eared headstalls and I don't like headstalls without a throatlatch. I feel like the horse could just rub it off. Plus, if one thing breaks, it is for sure going to fall off. But that's my personal preference.
∞•*˚ Βгįťţαňγ ˚*•∞
It is not enough to know how to ride; one must know how to fall.