What cards can be applied for and get approved through equifax only?

Why does it matter? Your credit report should be about the same for all of them.

That's not really true, I had fraud on my credit and was able to clean it up completely on one (equifax) but not the others yeat. I did have a BK two years ago and I have been able to get , run through Equifax only , sears, cap one, kays, lord and taylor, chevron and Cortrust mastercard .

That doesn't make you credit worthy... just overlooked.