[Zigûr]

Wikipedia's articles on Sauron and Morgoth actually contributed to my motivation towards contemplating this topic. Once upon a time (before I edited it actually) the article on Morgoth claimed that Sauron stayed loyal to Morgoth after the First Age but gave no evidence from Professor Tolkien's own writing to support this assertion. Indeed almost all of the evidence which I mentioned from "Myths Transformed" along with the note from Letter 183, both of which were kindly provided in detail by Boromir88, would suggest to me that after the War of Wrath Sauron was pursuing his own agenda. Morgoth had become, more or less, irrelevant beyond being a phantom from Sauron's personal history he could use to corrupt the Númenoreans and other Men he duped into perceiving him as a divinity.
While I believe that Sauron was indeed loyal to Morgoth in the First Age, and was motivated to this allegiance initially due to admiration for Morgoth's ambition and wilfulness and later because being Morgoth's second-in-command was a position of enormous authority he could never have achieved on his own given the circumstances of the First Age, I don't believe that loyalty persisted after Morgoth's exile. This quote from Morgoth's Ring seems pertinent:

While Morgoth still stood, Sauron did not seek his own supremacy, but worked and schemed for another, desiring the triumph of Melkor, whom in the beginning he had adored. He thus was often able to achieve things, first conceived by Melkor, which his master did not or could not complete in the furious haste of his malice.

I am noting here words and phrases including "while Morgoth still stood" and "whom in the beginning he had adored". The suggestion seems very much to me to be that Sauron's admiration for Morgoth had gradually eroded, probably due to Morgoth's increasing nihilism, and that after Morgoth's defeat Sauron was seeking his own supremacy.
I think by and large the confusion that his loyalty was more enduring seems to derive from a misinterpretation of the events of the Akallabêth. As for this "bringing Morgoth back" idea, I'm pretty sure that's just a silly notion I read somewhere with no basis whatsoever in the Professor's writing, and possibly derives from a desire some might have for Tolkien's world to be more like the heavily serialised Fantasy sagas of more modern times (the plot of A Darkness at Sethanon is something like that I believe).