The only ones I've seem consistently pull TU are Barclays and GEMB. Chase pulled ex for me, and I'm on the west coast. They told me that they use all 3 and couldnt confirm which one would be pulled until after it was done. Also many CUs tend to use TU, but you'd need to call them and find out which they pull.
Generally speaking, GECRB (Amazon, Walmart), Barclay's, BofA and occasionally credit unions (who don't use EQ) will pull TransUnion.
That being said, regional factors will also dictate whether Citi or Discover are EX or EQ (respectively) will be used, or both TU.
AMEX you can nearly bet your life will pull Experian. Occasionally, fax-ins due to freezes diverting the HP could render another report being pulled or viewed.
Capital One likes all three. In fact, they often pull three simultaneously. They'd pull your high school transcripts if possible.
The rest of your national issuers are a crapshoot as to what they'll pull. Again, it's usually a 'regional' thing.
It's not really a question of best. It's more a matter of what you're looking for and which rewards are best for you.