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.
Don't forget the images in Revelation:
And especially this one, which gives the Number of the Beast:
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.
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 separtion 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 as they are, the sight will evoke the reaction people have when the encounter say a snake, scorpion, etc.
In the bible, Satan or the Devil has been called the primeval serpent and dragon. This section has covered in a couple questions that highlighted the common threads and recurring themes among religions and peoples. We find dragons among the Chinese, and there is Quetzalcoatl a Mesoamerican deity whose name comes from the Nahuatl language and means "feathered serpent"; their depictions, terrifying.
What follows below is shared for knowledge:
In reply to the question WHAT IS A DEMON?, an exorcist Fr. Fortea says in it:
My commentary is we use terms as 'monster,' 'glazed-eyed/empty-soul,' 'repulsive,' 'evil-looking,', etc. to describe say evil people like serial killers, etc. Also, those in the Adult industry, ladies of the night [e.g. in dressing], etc, have a certain distinctive look about them.
Summary, demons and evil have a real [as God sees them] and distinctive appearance: terrifying to behold (hence the representation as serpents, dragons, goats, etc.), dark, ugly, repulsive, agitating, etc. everything the apposite of glory, good, beauty, welcoming, reassuring, etc.
Another description by by a seer at Fatima:
The first secret [shown by the Our Lady of Fátima] was a vision of hell, which Lúcia describes in her Third Memoir as cf. [The Message of Our Lady of Fatima]: