Most participants in these debates agree on the following:
That the Balrog in Moria had "wings" of some sort, or if you prefer, a "shadow" shaped like wings. Those "wings" were probably not made of flesh and blood, but rather of some sort of "dark emanation" or "palpable darkness".
Many of the most intense arguments seem to have resulted from different uses of the word "wing". In this consensus statement, the word "wing" is used only as a convenient symbol for the feature of the Balrog under discussion, without reference to any standard definition. In particular, the statement does not specify whether the "palpable darkness" always had a wing-like shape.
So what are the different definitions that people use? The Oxford English Dictionary divides its relevant definitions of the word "wing" into two groups. Group 1 includes definitions that for the most part refer to physical parts of a creature's body. For example "each of the organs of flight of any flying animal" (but broadened to include cases where similar organs are not used for flight, such as penguins' wings and even "the enlarged fins of flying fishes").
Group 2 includes definitions relating primarily to a thing's shape or position. For example, an appliance or appendage resembling or analogous to a wing in form or function", including an artificial apparatus attached to the human arms or shoulders" and "one of the planes of an aeroplane". Even broader, "a lateral part or appendage: in various connexions."
With a Group 2 definition of "wing", the question "Do Balrogs have wings?" is more or less trivial: Tolkien's description clearly fits (at least at the time described in the quotes below), as does the consensus statement above. With a Group 1 definition of "wing", the question is more interesting and the answer less obvious: it depends on the specific definition, and even then not everyone will agree.
The consensus statement above was not reached quickly, and even with a Group 1 definition some newcomers believe that a simple yes or no is clear from the description of the Balrog in "The Bridge of Khazad-dum". The debates generally begin as follows:
Those in favour of wings point out that when the Balrog steps onto the Bridge, "its wings were spread from wall to wall".
But those against point out that the word "wings" was first used in the phrase "the shadow about it reached out like two vast wings", and deduce that the word "wings" refers to the "shadow" itself rather than to a part of the Balrog's solid central body. (The "shadow" was seen when the Balrog first appeared: "it was like a great shadow, in the middle of which was a dark form", and it is likely the "cloud" in the phrase "It came to the edge of the fire and the light faded as if a cloud had bent over it".)
They also that claim that as the Balrog came closer to the Fellowship its uncertain appearance became clearer, so what first looked like a "shadow" or "cloud" and later "like wings" was finally recognized as "wings" once the Balrog was nearby. Those against wings do not agree with this interpretation.
Progress beyond this point is difficult, but again, most of those on both sides of the debate agree with the consensus statement. One substantial remaining point of disagreement is whether the "wings" always had a wing-like shape (supported in part by Tolkien's use of the direct phrase "its wings") or if their form was variable (supported in part by the "cloud" description), and there is no firm evidence known for either position.
