In my area, the western dressage riders seem to use reining saddles mostly. I audited a clinic and the clinician said that the typical equitation/western pleasure saddle puts riders in the wrong position, and reining saddles sit them in a good spot but also have more freedom in the fenders for better/more subtle leg aids.
I do see training saddles used as well, and one rider I know of uses a wade.
The local dressage club that offers western tests also allows the use of aussie, endurance, and plantation type saddles (like Orthoflex) for western dressage tests, but I don’t know if all clubs/rulebooks are the same in that regard.