How did dragons get their wings?
Modern day dragons are, in Western culture at least, imagined as huge, reptilian, fire breathing creatures with both legs and wings. However, as far as I can find out myself by research, dragons were developed from ancient mythical creatures that were more akin to gigantic serpents or worms. In ancient Germanic lore at least, they were called "worm", "orm", "wyrm" and later "lindworm". Now worms or snakes don't normally have legs or even wings. But up through the ages, these were added to their characteristics.
The epic poem of Beowulf was the first to describe a dragon/wyrm as having a fiery breath and bat-wings. Lindworms are often pictured in Romanesque art adorned with wings. Does anyone have a clue, at what stage, and under influence from what cultural tradition, did wings (and fiery breath) get added to the dragon as to arrive to our modern concept of these legendary beasts?
Are you asking about dragons in general, from absolutely any mythology, or exclusively from Germanic lore?
This question got me thinking "Why should dragons fly?" There is nothing in the Greek root for dragon that suggests flying. (The word is instead associated with vision, likely a reason the Python becomes associated with Delphi and Apollo.)
I thought about the famous Norse dragon Níðhöggr, who also doesn't seem to need to fly, as the creature dwells at the roots of the world tree which it reportedly likes to gnaw.
However, I found this interesting line at the end of the Völuspá:
The text of Beowulf predates the codification of the Poetic Edda, but the Norse material, but the religious poems that constitute the Poetic Edda certainly predate the text.
The dragons on the reconstructed mosaics of the Ishtar Gate certainly don't have wings, but as Gibet has noted, the dragon Mušḫuššu has been depicted with wings, and Tiamat, the primordial Babylonian goddess, is a type a type of dragon, and has been depicted with wings.
The association with fire seems to predate even Ancient Greece. See the discussion in the comments to this answer for some thoughts on the Greek relationship of dragons with fire.
"The dragons on the Ishtar Gate certainly don't have wings"... Certainly... on the reconstructed Ishtar Gate... But wings Mushussu got http://www.ezida.com/images/ninghi2 (the serpents "are" Ningishzida). Wings are quite common on Babylonian arts. Tiamat is almost systematically represented with wings (almost). And Marduk with 2 pairs. Those wings are feathered.
@Gibet + Duke have u considered if there is relationship between Mushussu and Qilin? Qilin is one of the 9 offsprings of dragon, according to Chinese myth.
@https://mythology.stackexchange.com/users/3960/mishu-%e7%b1%b3%e6%ae%8a If it exists any relation between Mushusshu and Qilin, Mushussu is the ancestor. But Tiamat would be a better ancesotr because its close proximity with the Indian Vritra (Indra vs Vritra being more or then Marduk vs Tiamat, or Ra vs Apopis, so you can see a form of propagation here). That said Mushussu was a god who was finally stuck with Marduk after being "absorbed"... and btw the dragon on the Ishtar gate is Mushussu (the "steed" of Marduk, the Merodach of the Bible).
Which one the ancestor an archeological's burden, my interest is in understanding the human mind/psyche.