At last the discussion of
Balrogs: Winged or not.
Personally I think Balrogs do have wings - however, they are not terribly functional in my mind.
Interesting to come to the conclusion that balrogs had wings with so little evidence. I would even say too simple, yet in saying that if they had wings that didn't function as intented (flying) is interesting.
Let's analyze this a little:
1. Balrogs had wings, (which there is no evidence in LOTR that they had any), and to go to the postion that the ones that they had don't function is intelligent and in a way convenient. Why, because you don't have to prove anything. In Lotr, which is canon
From Chapter 5: The
Bridge of Khazad-dûm
Something was coming up behind them. What it was could not be seen: it was like a great shadow, in the middle of which was a dark form, of man-shape maybe, yet greater; and a power and terror seemed to be in it and to go before it.
It came to the edge of the fire and the light faded as if a cloud had bent over it. Then with a rush it leaped across the fissure. The flames roared up to greet it, and wreathed about it; and a black smoke swirled in the air. Its streaming mane kindled, and blazed behind it. In its right hand was a blade like a stabbing tongue of fire; in its left it held a whip of many thongs.
'A
Balrog! A Balrog is come! '
Look at the way
Gandalf talked to the Fellowship in there:
'Over the bridge!' cried Gandalf, recalling his strength. `Fly! This is a foe beyond any of you. I must hold the narrow way. Fly!
Was Gandalf telling the fellowship to take out their wings and literaly fly (in the air) or to move out of there fast.

. Wait, there is the possiblity that they had wings, but they didn't function.
Later in the chapter:
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. It raised the whip, and the thongs whined and cracked. Fire came from its nostrils. But Gandalf stood firm.
I wonder why people don't quote this part entirely and only the part of the wings.

These quotes describe the balrog as a being of "shadow" and its "shadow" reached out like two vast wings, not that it had 2 wings.
The Balrog made no answer. The fire in it seemed to die, but the darkness grew. It stepped forward slowly on to the bridge, and suddenly it drew itself up to a great height, and its wings were spread from wall to wall; but still Gandalf could be seen, glimmering in the gloom; he seemed small, and altogether alone: grey and bent, like a wizened tree before the onset of a storm.
It says that the darkness grew and so its "wings". These are of course not real wings.
With a terrible cry the Balrog fell forward, and its shadow plunged down and vanished. But even as it fell it swung its whip, and the thongs lashed and curled about the wizard's knees, dragging him to the brink. He staggered and fell, grasped vainly at the stone, and slid into the abyss. 'Fly, you fools! ' he cried, and was gone.
It says that the shadow vanished, and while it did, there is no mention of its "wings of darkness". If he had any wings, why aren't they mentioned here?
And again, Gandalf tells the company to get out of there by using the word fly, which i don't think means to intend for them to go "flying" in the air.
From LOTR:
The White Rider
'Yet it has a bottom, beyond light and knowledge,' said Gandalf. 'Thither I came at last, to the uttermost foundations of stone. He was with me still. His fire was quenched, but now he was a thing of slime, stronger than a strangling snake.
At this time, did he had "useless wings of slime" then?
'There upon
Celebdil was a lonely window in the snow, and before it lay a narrow space, a dizzy eyrie above the mists of the world. The sun shone fiercely there, but all below was wrapped in cloud. Out he sprang, and even as I came behind, he burst into new flame. There was none to see, or perhaps in after
ages songs would still be sung of the
Battle of the Peak.' Suddenly Gandalf laughed. 'But what would they say in song? Those that looked up from afar thought that the mountain was crowned with storm. Thunder they heard, and lightning, they said, smote upon Celebdil, and leaped back broken into tongues of fire. Is not that enough? A great smoke rose about us, vapour and steam. Ice fell like rain. I threw down my enemy, and he fell from the high place and broke the mountain-side where he smote it in his ruin.
Any mention of wings anywhere? Even useless wings?

Let's look at other mentions of the word fly in LOTR
From LOTR:
Helm's Deep
'Would that day was here and we might ride down upon them like a storm out of the
mountains!' said
Aragorn. 'It grieves me to fly before them.'
'We need not fly much further,' said
Éomer. 'Not far ahead now lies
Helm's Dike, an ancient trench and rampart scored across the coomb, two furlongs below Helm's Gate. There we can turn and give battle.'
If I remember correctly, Aragorn couldn't fly, could he?
From the Published Silmarillion: Of the
Flight of the Noldor
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.
Wings anywhere? Or even mention the word fly?
But to be fair, we have in
Morgoth's Ring
From Morgoth's Ring: Later
Quenta Silmarillion:
Far beneath the halls of Angband, in vaults to which the Valar in the haste of their assault had not descended, the Balrogs lurked still, awaiting ever the return of their lord. Swiftly they arose, and they passed with winged speed over Hithlum, and they came to Lammoth as a tempest of fire.
We have here the mention of the phrase "winged speed" that means either that they moved very fast or that they had wings could fly, or simply that the wings that they had made them move more "rapidly".

There is another phrase with the word "winged":
Therefore
Ulmo taught them the craft of ship-building; and Ossë, submitting to Ulmo, brought them as his farewell gift the strong-winged swans.
Swans definitely had wings, so does that means that the balrogs had wings too?
I would conclude that based on the canonical evidence in the LOTR, you cannot say that Balrogs had wings or useless wings, because there is no proof.
If you decide to use the Later Quenta as the source of information, you have to remember that it's not canon, because it was not published by JRRT. You could say that they either had: wings or useless wings.
In conclusion, if you were to say that the Balrogs had wings or useless wings, the proof you have is in the Later Quenta, not in LOTR nor the Published Silmarillion.