This question is usually asked in relation to the debate about the wings of Balrogs, and specifically whether those wings existed at all. For some background on this topic, see the main Encyclopedia's entry for Balrogs. Sources are indexed by bracketed indexes in the text - a detailed list is given at the end of the article.
'Then something came into the chamber - I felt it through the door, and the orcs themselves were afraid and fell silent.'(3) These are the words of Gandalf describing his first encounter with the Balrog in Moria, and they raise a point that can be used to argue against the idea of the Balrog having wings. We have a lot of information about this chamber - the Chamber of Mazarbul - and all of the available evidence points to its having a narrow entrance. The argument is described in much fuller form in the Balrogs entry itself, but in essence it is this: if the Balrog really was a being with wings that could span a cavernous hall, it should not have been able to get through this narrow door.
A common counter to this point is a suggestion that the Balrog actually changed shape between the two events, effectively growing a pair of wings between the incident in the Chamber and meeting Gandalf on the Bridge. This may seem a strange argument at first, but in fact there's nothing inherently implausible about it, and it's presented so regularly that this article tries to address some of the issues it raises.
There are two realistic ways that the Balrog could have achieved this shape-shifting feat. The first relates to its 'shadow', a mysterious region of darkness that the Balrog seems to carry with it. The suggestion here is that the Balrog didn't change shape itself, but rather its shadow took different forms in the two different situations. This is quite well supported by the text: Gandalf reported that, looking back into the Chamber of Mazarbul, 'Something dark as a cloud was blocking out all the light inside...' (3), and later at the Bridge of Khazad-dûm, '...the shadow about it [the Balrog] reached out like two vast wings...' (3). This view is also supported by The Silmarillion: '...[the Balrogs'] hearts were of fire, but they were cloaked in darkness...' (2). This seems a perfectly reasonable interpretation, then, and it leads to the conclusion that the Balrog's wings were of a cloudy, metaphorical sort. This is not, however, the most common form of the argument - much more popular is a proposition that allows the Balrog to have real, physical wings, and it's that idea that we address in the rest of this article.
This second approach is based on a description of the Ainur in Ainulindalë, where we're told that the Valar, at least, '...may clothe themselves in their own thought, made visible in forms of majesty and dread.' (1) We know that this power was shared by the lesser beings of the same kind as the Valar, the Maiar, and Balrogs were beings of this sort. Indeed, there is even an explicit account of a Maia growing wings to escape from a battle - Sauron took on a flying form after his encounter with Huan the Hound. According to this reading, the Balrog had no wings in the Chamber of Mazarbul, nor when it was first encountered at the Bridge, but at the point where Tolkien writes '...suddenly it drew itself up to a great height, and its wings were spread from wall to wall...' (3) he is describing the Balrog willing a pair of mighty wings into existence. This is a consistent approach, and it does explain how the Balrog could travel through a narrow doorway at one point, and later display a huge pair of wings. Whether this was Tolkien's own intention, though, remains open to question.
To complicate matters, it isn't clear that Tolkien considered Balrogs to be Maiar at the time he wrote this passage. In his original conception, he saw Balrogs as creatures created by Morgoth, and not Maiar at all. The first references to their later origin as Maiar appear in texts from the 1950's, dated ten years or more after the Balrog descriptions in The Lord of the Rings were written. It's impossible to know for sure whether the Balrog of Khazad-dûm, as Tolkien imagined it at the time, was a Maia or not. Indeed, the fact that it is twice referred to as a 'Balrog of Morgoth' (by Legolas in (4), and also in (5)) hints that Tolkien imagined it as Morgoth's creation, rather than his ally.
Perhaps the main objection to the shape-changing view, assuming that Balrog actually was a Maia in Tolkien's imagination, is that it relies on information from The Silmarillion, information that Tolkien knew would not be available to his readers. We know now that Balrogs are Maiar, and that Maiar generally have the ability to change shape, but there's nothing in The Lord of the Rings itself to suggest any of this. If the idea of the shape-changing Balrog is correct, then we have to assume that Tolkien meant his readers to guess that the Balrog could grow wings at will, without any background information about its abilities. This seems curiously imprecise, particularly for a writer of Tolkien's fastidious nature.
If we do imagine the Balrog creating wings for itself, then a further objection arises - the fact that these wings are utterly useless. Moments after creating them, the Balrog is unable to stop itself from falling into a deep chasm. In the broader debate, this point is typically answered by suggesting that the Balrog's wings were too wide to deploy properly, or that it was unable to make use of them while in the act of falling. If the Balrog has created its own wings, though, these points simply don't apply - the Balrog could have created perfectly usable wings if it had wished to do so. For that matter, if the Balrog really could change his shape, the chasm should have prevented no difficulties at all: he could simply have turned himself into a flock of birds, or grown a parachute! On balance, the idea of a shape-shifting Balrog seems difficult to reconcile with the events described in The Lord of the Rings.
Even if we accept it as a realistic possibility, though, the idea doesn't seem to advance the more general 'wing' argument very far. That's because there is only one reason to think that the Balrog could change shape: to explain how it could have had real wings at the edge of the chasm. In other words, we can't work out whether the Balrog changed shape unless we already know that it did have wings, because if it didn't, there's no evidence for its shape-shifting at all. That leads us into a circular argument: 'if the Balrog had wings, then it must have changed shape; if the Balrog changed shape, then it must have had wings'. What's missing here is some concrete evidence for the shape-shifting theory, some statement in the text that can only possibly be interpreted as describing a shape-changing, wing-growing Balrog. Without that, the idea remains in the area of speculation - an unlikely turn of events, but not quite impossible.
||Quenta Silmarillion 3, Of the Coming of the Elves and the Captivity of Melkor
||The Fellowship of the Ring II 5, The Bridge of Khazad-dûm
||The Fellowship of the Ring II 7, The Mirror of Galadriel
||The Lord of the Rings Appendix A III, Durin's Folk