Angels are normally portrayed in illustrations as having wings, but the Bible normally presents them as appearing like men (and so without wings). For example, three angels appeared to Abraham and at first he thought they were men (Genesis 18:2). Similarly, when an angel appeared to Samson’s parents, they thought he was a “man of God” (Judges 13:9-10). Hebrews 13:2 says that some people have “entertained angels unawares”, indicating that they appear as ordinary human beings.
Sometimes angels appear in white. Examples are John 20:12 and Acts 1:10. But wings are never mentioned.
There is one passage which describes the angel in “swift flight” (Daniel 9:21), but again no wings are mentioned. And there is no reason to think that a spirit being would need wings to fly.
There are two angel-like creatures that do have wings, and this may be the origin of the idea that angels have wings. Cherubim (Exodus 25:20; Ezekiel 10) and Seraphim (Isaiah 6) are both described as having wings. However, these are not angels. Cherubim are living creatures that act as guards to God’s dwelling place. Apart from Genesis 3:24, the other references to cherubim are all to images or visions. Seraphim means “flaming creatures” and are only mentioned in Isaiah 6 — a vision of Isaiah about God’s dwelling place.