I'm thinking about joining a local gun club that hosts SASS (Single Action Shooting Society) matches.
Mostly to use the shooting range for fun and practice, but also was thinking about shooting competitively.
What hand gun (brand / caliber / etc) would be the best or most popular. What do they use and/or allow?

Ruger Vaquero is a popular choice

So single action, .45 Colt.
I was also wondering if they were using black powder cartrige type guns?
I wish I knew a local member to inquire. But I don't.

Take a look at the USFA single action "Rodeo" model. It is a fine American made(old Colt plant) revolver that closely mimics the original Colt Peacemaker at a substantially less price. As far as cartridge selection, I would go with the old workhorse---45 Colt. You can shot BP if you want, and there are plenty of good bullets and loads, both smokeless and BP, for the 45 Colt. Tom.

Go to an event and see what the folks are using. Good chance most will let you handle them and be happy to answer questions. Going with the most common isn't a must anyways. I have a set of .32 H&R mag Single Sixes and a 1894 Marlin in the same chambering.
The events I've attended, Ruger was by far the prdominant brand. Again, all depends on what you want. What class you'd shoot in, etc...

There is no doubt that the Colt SAA Army is top shelf, but just starting out and not knowing if you are going to get hooked into this sport, I would go for the Ruger Vaquero. Well made handgun and very accurate and reliable.