Winged beings in heavenly visions
The popular image of a winged angel has real roots in Scripture. When Isaiah saw the Lord in the temple, he saw seraphim, heavenly beings with six wings. They covered themselves in God's presence and flew as they served him (Isaiah 6:2).
Cherubim are also connected with wings. On the ark of the covenant, God commanded Moses to make cherubim whose wings stretched above the mercy seat (Exodus 25:20). Ezekiel's vision likewise describes living creatures with four wings (Ezekiel 1:6), and later identifies those living creatures as cherubim (Ezekiel 10:15).
Above it stood the seraphims: each one had six wings; with twain he covered his face, and with twain he covered his feet, and with twain he did fly.
Angels often appear as people
When angels meet people on earth, the Bible usually doesn't mention wings. Two angels arrived at Lot's home in Sodom, but the people of the city spoke of them as men (Genesis 19:1-5). Their appearance was human enough that others did not immediately recognise who they were.
Hebrews says that people have sometimes welcomed angels without realising it (Hebrews 13:2). That doesn't mean angels are human. It shows that God can send them in an ordinary-looking form when they carry out his purposes.
Be not forgetful to entertain strangers: for thereby some have entertained angels unawares.
Flying does not settle every detail
Revelation describes an angel flying through the midst of heaven while proclaiming God's message (Revelation 14:6). The passage clearly pictures flight, but it doesn't specifically mention wings. So it would be going beyond the text to decide from this verse exactly how that angel moved.
The Bible uses different names for heavenly beings, including angels, seraphim, and cherubim. It doesn't give a complete physical description of every one. Visions show genuine heavenly realities, but they don't require us to assume that every heavenly messenger looks alike.
The main thing Scripture shows
So, do angels have wings? Some heavenly beings certainly do in biblical visions. But Scripture never says every angel has wings, and its accounts of angels visiting people often present them as men rather than as winged figures.
The Bible draws attention less to their appearance than to their service. Angels are God's ministers, sent to serve those who will inherit salvation (Hebrews 1:14). Their purpose is not to become the centre of our attention, but to point to the God they faithfully obey.
Are they not all ministering spirits, sent forth to minister for them who shall be heirs of salvation?