The infamous debate is about whether or not the Balrogs have wings and whether or not they can fly. It's quite simple to understand, too: initially it is very clear Tolkien uses the word 'wings' figuratively; what continues the debate is whether the second use is meant to be literal or figurative. At first glance it seems to many that it's to be literal. Tolkien however takes liberty with figurative senses for many words: 'fly' is another example; very often he uses it figuratively meaning 'flee'. But the fact it appears Balrogs have wings combined with the liberal use of the word 'fly' is quite ambiguous to many people (even though there isn't a reference to the Balrog where the word 'fly' is used it is definitely used in the chapter).

As a literal thinker who also has an amazing imagination I want to discuss this from a different viewpoint. What do I hope to accomplish in writing this? I can say it's probably not what most would think: I am in no way making any attempt whatsoever to change the opinions of anybody; not only are we all prone to confirmation bias but it also serves no purpose whatsoever trying to change the opinions of others of a novel of any genre and least of all fantasy. What would I gain by changing the way someone uses their imagination? What would it gain me to potentially remove or lessen the joy and magic of reading? The joy of reading is to make use of the magic of words in whatever ways makes [you] most happy. That is what imagination allows and it is a wondrous gift that nobody should be deprived of. Instead I am only offering my thoughts for my enjoyment of analysing it as well as the enjoyment of anyone else who might find the subject, or my take on it, interesting.

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.

And that leads me to the next section: the debate itself.

In order to analyse this we must first find the source of the debate. More specifically we must identify the passages fuelling the debate. In the chapter The Bridge of Khazad-dûm we have the following passages; I will dissect each passage separately before tying them all together:

The dark figure streaming with fire raced towards them. The orcs yelled and poured over the stone gangways. Then Boromir raised his horn and blew. Loud the challenge rang and bellowed, like the shout of many throats under the cavernous roof. For a moment the orcs quailed and the fiery shadow halted. Then the echoes died as suddenly as a flame blown out by a dark wind, and the enemy advanced again.

'Over the bridge!' cried Gandalf, recalling his strength. 'Fly! This is a foe beyond any of you. I must hold the narrow way. Fly!' Aragorn and Boromir did not heed the command, but still held their ground, side by side, behind Gandalf at the far end of the bridge. The others halted just within the doorway at the hall's end, and turned, unable to leave their leader to face the enemy alone.

It's pretty clear that the Fellowship cannot themselves fly; the use of the word 'fly' most certainly does not mean for them to do anything but flee. You'll find, too, that one of the definitions of 'fly' is: informal Depart hastily. ‘I must fly!’

This very fact leads to the conclusion that Tolkien does make figurative use of the word; in fact this isn't even remotely the only times he uses the word 'fly' in this sense. But there is another word involved in the debate: wings. That leads to the next passage.

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.

[...]

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.

And that is the source of the debate: it seems ambiguous because initially it's clearly figurative; but then the next use it appears to forego the previous figurative use: if it no longer has the word like then surely it must now not be figurative.

But is it that simple? If someone makes an analogy in a discussion and continues that discussion by referencing that analogy without directly stating it is a analogy, is it no longer an analogy? Not only is it rather redundant but it also makes the assumption that the audience has an extremely short attention span. That isn't to say those who go from figurative to literal in a moment have short attention spans; it is to say that one shouldn't have to specify that each reference to an analogy just made is in fact still related to the analogy; on the contrary it might be said one would have to specify they no longer refer to the analogy: otherwise it is a problem of continuity.

But let's assume that there isn't a continuity problem. Let's instead analyse it as a literal thinker would. A literal thinker typically takes things literally as the name might suggest. But that doesn't mean a literal thinker never knows when something is meant figurative and it doesn't mean a literal thinker can never be figurative.

What it does mean is simply this: if I as a literal thinker understand the author is being figuratively when describing something, for instance the Balrog wings, then because I interpret things literally I now must interpret future references to that same thing as it was originally meant: figuratively. That's all there is to that: if we interpret something a specific way then if we are being literal we will continue to interpret it that same way.

We already know Tolkien uses the word 'fly' figuratively; but if we follow the logic I just explained then because it was never applied to the Balrog it doesn't apply: figuratively or literally.

But for the purpose of the discussion here, we'll assume in fact Balrogs do have wings. One of the arguments is that flying creatures don't necessarily start flying when falling. I won't deny that; neither will I deny that the wings could be damaged (though certainly Tolkien described no such thing and in fact doesn't mention wings again at all when referring to the Balrog known as Durin's Bane). At the same time though as I've already mentioned whenever a Balrog is slain the opponent as well as itself falls. This happened to two Balrogs slain in the fall of the city Gondolin.

There is something else though: a flying creature when the Fellowship are on the River Anduin:

'Elbereth Gilthoniel!' sighed Legolas as he looked up. Even as he did so, a dark shape, like a cloud and yet not a cloud, for it moved far more swiftly, came out of the blackness in the South, and sped towards the Company, blotting out all light as it approached. Soon it appeared as a great winged creature, blacker than the pits in the night. Fierce voices rose up to greet it from across the water. Frodo felt a sudden chill running through him and clutching at his heart; there was a deadly cold, like the memory of an old wound, in his shoulder. He crouched down, as if to hide.

Suddenly the great bow of Lórien sang. Shrill went the arrow from the elven-string. Frodo looked up. Almost above him the winged shape swerved. There was a harsh croaking scream, as it fell out of the air, vanishing down into the gloom of the eastern shore. The sky was clean again. There was a tumult of many voices far away, cursing and wailing in the darkness, and then silence. Neither shaft nor cry came again from the east that night.

[...]

'But who can say what it hit?' said Legolas.

'I cannot,' said Gimli. 'But I am glad that the shadow came no nearer. I liked it not at all. Too much it reminded me of the shadow in Moria - the shadow of the Balrog,' he ended in a whisper.

'It was not a Balrog,' said Frodo, still shivering with the chill that had come upon him. 'It was something colder. I think it was -' Then he paused and fell silent.

'What do you think?' asked Boromir eagerly, leaning from his boat, as if he was trying to catch a glimpse of Frodo's face.

'I think - No, I will not say,' answered Frodo. 'Whatever it was, its fall has dismayed our enemies.'

It's pretty clear to me Frodo knew what it was: a Nazgûl. Note how he clutches his heart as if a memory of a wound in his shoulder; that itself is a good indication he knows for it was his heart that the blade of the Morgul-knife was trying to get to from his shoulder; this is not the only time this happens. Whether he knew this or no the fact it was a Nazgûl is confirmed by Gandalf in The Two Towers:

'The Winged Messenger!' cried Legolas. 'I shot at him with the bow of Galadriel above Sarn Gebir, and I felled him from the sky. He filled us all with fear. What new terror is this?'

'One that you cannot slay with arrows,' said Gandalf. 'You only slew his steed. It was a good deed; but the Rider was soon horsed again. For he was a Nazgûl, one of the Nine, who ride now upon winged steeds.

So it was not the Balrog nor was it any Balrog; the only Balrog encountered in The Lord of the Rings is Durin's Bane and Gandalf destroyed it just as he himself was destroyed (but life returns to Gandalf). Incidentally, Christopher Tolkien suggests in The Return of the Shadow (or was it the next volume in the History of The Lord of the Rings? The Treason of Isengard; I can't recall but he certainly does refer to this specifically) that the very first example of Winged Nazgûl is the shape the Fellowship encounter in the sky (that they feel) prior to entering the Mines of Moria.

The question finally then is can the Balrogs fly? While it's pretty clear they do not have wings they most certainly are powerful. But even assuming they do have wings there is not a single reference to Balrogs flying; this isn't conclusive evidence that they cannot fly but it's all we have: there is no evidence to them flying and that's all there is to it.