The description of the Balrog in Moria from "The Bridge of Khazad-dûm", in the fifth chapter of The Fellowship of the Ring raised the question of whether Balrogs were winged. There are two references in this chapter:
His enemy halted again, facing him, and the shadow about it reached out like two vast wings.
...suddenly it drew itself up to a great height, and its wings were spread from wall to wall...
Tolkien's language leaves some room for speculation. The first quote seems to describe a Balrog covered in shadow that appears winglike, and is later 'spread.' However, the second quoted passage seems to indicate that this Balrog had actual wings that were spread from wall to wall.
Arguments for Balrog wings
The most common reference for winged Balrogs is the 'vast wings' language in "The Bridge of Khazad-dûm". The plain language here explicitly refers to 'wings.' Another Tolkien reference may be taken as evidence of Balrog wings:
This is either a metaphor for moving very quickly, or a literal reference to physical wings - it is this issue over the verb 'to fly' which is mostly at the heart of the Balrog debate. Another, oft-quoted, example from The Appendices:
Thus they roused from sleep a thing of terror that, flying from Thangorodrim, had lain hidden at the foundations of the earth since the coming of the Host of the West: a Balrog of Morgoth
— Appendix A, Durin's Folk
Arguments against Balrog wings
On the other hand, the explicit reference to "wings of shadow" leaves open the interpretation that Balrogs had no physical wings. Balrogs were never exactly described as flying in any of Tolkien's works, including the "winged speed" language quoted above. Furthermore, at least two Balrogs fell to their ruin, apparently wingless:
Many are the songs that have been sung of the duel of Glorfindel with the Balrog upon a pinnacle of rock in that high place; and both fell to ruin in the abyss.
—Quenta Silmarillion, Chapter 23, "Of Tuor and the Fall of Gondolin"
Some think the strongest objection is the simplest: that taking references like the second statement seriously mean that all lines must be taken literally. For example, shortly before the Balrog's appearance, "Gandalf came flying down the steps and fell to the ground in the midst of the Company."
Other versions of the legendarium
The first mention of Balrogs in the legendarium is in "The Fall of Gondolin" of The Book of Lost Tales, in which they are lesser beings and counted by hundreds. In their description there is no mention of wings.:169 There is also an explanation of how Melko did not know the secret of flying, so he caught many Eagles to squeeze them this knowledge, but they did not tell him anything, so he cut their wings and made a pair of wings for him.:193 This would not be necessary if Melko's Balrogs already had the ability to fly; therefore, in their first conception, Balrogs did not have wings.