Many people have seen the Classical depiction of Satan in paintings and statues, but what does scripture say about what Satan looks like?
Apparently, anything he likes, including an angel:
Assuming Satan is an angel, then he can probably take on any appearance, at least that of a human:
And possibly a snake (though some Christians do not take this passage literally)
More information can be found on the Christian demonology Wikipedia article.
Truth's own testimony
A look at the returns on biblical search on Satan shows that there is no biblical description of how Satan looks like, and therefore, the answer that follows will be what we can deduce Satan looks like from the Bible.
Since Satan goes to great lengths to disguise himself as an angel of light, he really must be anything but what an angel of light looks like.
God differentiates evil from Good that he IS. Just as he separated light from darkness.
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Only God is Good [cf. Mk 10:18] and he alone is creator. Good and evil belong to the knowledge of God [cf. Gn 3:5], therefore it is he who determines what evil should look like since he is the only Good. Hence the separation of what is clean and unclean, what is an abomination or not as in Lv Ch 11 and Dt Ch 14. For example:
We know most people's reaction when they see a snake, or a centipede, or a scorpion.
Another illustrative example in Rv 18:2
Concluding, in general, should one see demons - Satan being their leader - as they really are, the sight will more likely evoke a worse reaction than people have when the encounter say a snake, scorpion, etc., consistent with the biblical description of Satan or the Devil as the primeval serpent and dragon. Since God is beauty and attractive, Satan must appear not-comely and repulsive, and Since God is Light and his children stars and light, Satan must be/appear dark. Finally since Satan disguises himself as an angel of light, he really must be anything but what an angel of light looks like.
The Bible says that Lucifer( Satan's former name)was a cherub(Ezekiel 28:14). Cherubs are not cute little chubby angels but are in truth actually pretty frightening to look at. Cherubs are twenty cubits long, had four faces(of an ox, lion, eagle and man), an enormous wingspan, huge flaming swords and are covered in eyes according to Ezekiel and Isaiah's accounts. We can only assume that Satan looks like one of these.
It should be noted that as an angel, aside from a stint in fleshly imagery in the West, Satan like other angels is considered to not actually have a 'true appearance' at all. He is a bodiless intelligence and is thus, invisible.
It is highly misleading to try to tack down his appearance, like that of any angel or demon, because what they are is not primarily bodily and thus, passively or objectively visible, but intelligible and intelligent, and subtle. Any image they assume can be considered a 'vision of angels', since they have no actual appearance. How they appear is based on how they wish to be intelligible to us.
If you can stomach it, you may wish to consider On The Divine Hierarchy by St. Denys.
Satan can appear as an angel of light and frankly, as alluded to above, whatever suits him and God permits.
Don't forget the images in Revelation:
And especially this one, which gives the Number of the Beast: