Twas all up to the Balrogs, they were Valaraukar - Maia spirits - and could all take on the form they wanted, and add any attributes to that form to their heart's content.
For instance, from chapter 9 of the Silmarillion :
Deep in forgotten places that cry was heard. Far beneath the rained halls of Angband, in vaults to which the Valar in the haste of their assault had not descended, Balrogs lurked still, awaiting ever the return of their Lord; and now swiftly they arose, and passing over Hithlum they came to Lammoth as a tempest of fire.
I think those Balrogs were winged, as i can't see how else they would've made it to Lammoth in time to save their master from Ungoliant's tender embrace. Also the fact that "passing over Hithlum" is mentioned, which to me indicates that flying was involved.
Other Balrogs though, were not winged. Remember that Glorfindel fought a Balrog in Gondolin and threw it down to its doom : if that Balrog was winged, wouldn't it have saved itself by using its wings ? Of course, the Balrog possessing wings doesn't necessarily mean it can fly, but why would it have wings then ? Another possibility is that the Balrog's wings were destroyed in the battle.
All the confusion of course, started with the following quote from The Bridge of Khazad-Dûm :
The Balrog reached the bridge. Gandalf stood in the middle of the span, leaning on the staff in his left hand, but in his other hand Glamdring gleamed, cold and white. His enemy halted again, facing him, and the shadow about it reached out like two vast wings.
Whilst JRRT only used a comparison, some ppl choose to take this literally.
Also note that Gandalf killed Durin's Bane by throwing it to its doom, which would mean that if the Balrog did have wings, it couldn't fly. Of course, it would be rather unlogical for a Balrog to have wings but not being able to fly : this would be inefficient, hence i think if Balrogs had wings, that they could indeed fly.
So the conclusion is that imo Durin's Bane was not winged.