Well
creatures that can fly aren't always flying.
As more and more someone who sticks strictly to the canon, I can only speak about the
Moria Balrog, and the fact that he fell into the chasm doesn't necessarily mean that he can't actually fly.
The Balrog basically means to chop
Gandalf into kebab sized pieces and is then blindsided by Gandalf using his staff - 'A blinding sheet of
white flame sprang up' (perhaps the Flame of
Arnor?); once he falls into the abyss he manages to drag Gandalf in with him. From then on the two are basically fighting hand to hand, entwined with each other, so the Balrog can't use his wings.
This is,of course, speculation, but there is another statement by
Tolkien which proves beyond doubt that the Moria Balrog does indeed have wings and can indeed fly:
'The
Dwarves delved deep at that time...thus they roused from sleep a thing of terror that, flying from
Thangorodrim, had lain hidden at the foundations of the earth...a
Balrog of Morgoth'. (Appendix A
Durin's Folk)
Open and shut case I reckon.