Sauron, Morgoth's first and foremost lieutenant, had a fair form which was lost only when Ilúvatar pulled Númenor down into the sea with Sauron's fair form trapped within it, such that Sauron's spirit was forced to abandon it. It seems to be suggested that all Ainur possess one.
Being somewhat evil since the beginning, does Morgoth possess a fair form at all? Does he still possess it?
We're told that, during the Years of the Trees, Morgoth (before he earned that name) walked among the Elves and tried to sway them against the Valar:
Now in his heart Melkor most hated the Eldar, both because they were fair and joyful and because in them he saw the reason for the arising of the Valar, and his own downfall. Therefore all the more did he feign love for them and seek their friendship, and he offered them the service of his lore and labour in any great deed that they would do.
The Silmarillion III Quenta Silmarillion Chapter 6: "Of Fëanor and the Unchaining of Melkor"
It must be supposed that he wasn't doing this in his "Great and Terrible" form, for obvious reasons. Moreover, we're told explicitly that he used a fair-seeming form when he went to Fëanor at Formenos (emphasis mine):
Fëanor's heart was still bitter at his humiliation before Mandos, and he looked at Melkor in silence, pondering if indeed he might yet trust him so far as to aid him in his flight. And Melkor, seeing that Fëanor wavered, and knowing that the Silmarils held his heart in thrall, said at the last: 'Here is a strong place, and well guarded; but think not that the Silmarils will lie safe in any treasury within the realm of the Valar!'
But his cunning overreached his aim; his words touched too deep, and awoke a fire more fierce than he designed; and Fëanor looked upon Melkor with eyes that burned through his fair semblance and pierced the cloaks of his mind, perceiving there his fierce lust for the Silmarils.
The Silmarillion III Quenta Silmarillion Chapter 7: "Of the Silmarils and the Unrest of the Noldor"
However, as Travis Christian points out, he lost the ability to use this (or any other) form after destroying the Two Trees.
Not anymore.
I don't have a quote for Morgoth's original form, or any before his rebellion, but we know that by the time of the First Age he was limited to the form that we usually see portrayed in artwork.
After he and Ungoliant destroy the Two Trees, Of the Darkening of Valinor says this:
Now Melkor came to Avathar and sought her out; and he put on again the form that he had worn as the tyrant of Utumno: a dark Lord, tall and terrible. In that form he remained ever after.
And an explanation by Tolkien in Morgoth's Ring:
To gain domination over Arda, Morgoth had let most of his being pass into the physical constituents of the Earth – hence all things that were born on Earth and lived on and by it, beasts or plants or incarnate spirits, were liable to be 'stained'. Morgoth at the time of the War of the Jewels had become permanently 'incarnate'.
But having been created as one the Valar who could and did change forms at will, presumably Morgoth (or rather, Melkor) would have taken one or more "fair forms" before this time.
