At AMHR breed shows, I'd say 100% of the people use the Arab type halter for the halter classes. Most people switch to a stock type halter for showmanship, obstacle, hunter, and jumper. At 4H over half the kids use a stock halter for everything.
We just went to an Open Horse Show today and we showed Lance in the "English Halter" division with the Arabs and Morgans, and we fit right in with our cable halter. Now, if my mare Spice had come, I would have shown her in the "Western Halter" division in her stock halter--she is a miniature Paint!!
At registry shows, for halter, model and color, you will hardly see anyone showing in a western style halter. People usually use arab / cable halters for that, HOWEVER, at local, open shows, you might be better off with a western show halter if that's what most of the other people (showing "big" horses) are using.
At registry shows for performance, most people prefer a western style halter. You can get some nice, thin strip kind from Ozark Mountain. The ones you can find on eBay are kind of thick and bulky looking on a mini's head.
For a halter class I would never use a western style halter no matter what "style" horse I was showing. They are too big, clunky and hide the head while making it look bigger than it is. Manufacturers of western style halters still haven't figured out that they need to make those leather straps MUCH narrower. Think about it...the straps on a quarter horse halter are 1" wide...and yet they make the straps on a little mini halter 3/4" wide which in my opinion looks ridiculous.
The only time I would even consider a western style would be for jumping (that is, if my knees weren't shot and I could run!) because the chain on a cable or "Arab" style halter might make a horse stop. That's why a lot of people actually use a barn halter for jumping.
If you are entering an AMHA or AMHR show, as opposed to an open show, the style of halter you use depends on the class(es) you are entered in. For halter, no matter what style of horse you have, you will want to use an arab style halter. For obstacle, showmanship and hunter/jumper, most use a QH style, but some people do use arab style. If you're wanting to get a nice QH halter that's not too big, they do exist, but you have to look around. I have one that is very dainty--the straps are certainly not anywhere near 3/4 inch wide. Good Luck!