It's about 2.5 years old (so very out of date, I'm sure), but this article from
Deeper Blue ranks PADI at about 60-75% of diver certifications (I'm assuming that only means OW), followed by SSI, having recently knocked NAUI out of 2nd. The top five are rounded out by RAID and BSAC. In the year 2001, in the US, 98.2% of divers are trained by PADI, SSI, or NAUI and the rest are slit up amongst 20 different agencies.
I did my OW with PADI and everything since with SSI. PADI required everyone to purchase printed materials (in 2013) individually and was more strict about the instructors following a script. Ours went a little off script one morning to teach us about BP/W, but otherwise you had to do skills in a particular order. SSI does all of the classroom training online followed by in class and there is nothing to buy and the instructors get a little more leeway in what order they teach each skill and how much time to dedicate to it. SSI also allows instructors to do combo dives, like a deep dive to a wreck can count for both classes, whereas I'm pretty sure PADI frowns on that. At the end of the day, the instructor matters a lot more than the agency, though. Some instructors will sign off on mastery of a skill if you do it once, others will want to see real mastery; some instructors will coach you and train you, others think they need to be a drill sergeant and just yell at you.
SSI does not require a certain brand, but they do require that you train in affiliation with a shop. If you shop is partnered with a certain brand or two (mine is SP and Atomic), instructors will almost certainly be required to wear that gear, usually from the shop's rental fleet, while training. We have several instructors who are both SSI and PADI instructors. They wear different gear if they are training independently (as PADI) than they wear if they are doing a shop class (with SSI).