Some great balrogs probably had wings, but not all of them.

No, balrogs definitely, clearly, authoritatively, do not have wings. This is extremely clear in the text.

They're always described as running and climbing, and Tolkien makes a big deal about the fact that Melkor does not have any winged support at all until the winged dragons are revealed at the very end.

Also, the fact that more than one balrog meets its end by plummeting off of a cliff is also highly suggestive.

The big reason people think they might or should have wings is because of the description Tolkien gives at the Bridge of Khazad Dum, where he states the balrog steps out, shadows spreads around it like two vast wings. He uses a simile, comparing. But it's just darkness. The balrog is so powerful that the darkness itself spreads around it, and so, in order to try to convey visually what that looks like, he compares the darkness to two wings.