As much as it pains me to deal with any mention of the show that shall not be named (similar to Elrond's expression here)...

Somewhat. At the start of "Of the Rings of Power and the Third Age" we find

When Thangorodrim was broken and Morgoth overthrown, Sauron put on his fair form again and did obeisance to Eonwe, the herald of Manwe, and abjured all his evil deeds. And some hold that this was not at first falsely done, but that Sauron in truth repented, if only out of fear, being dismayed by the fall of Morgoth and the great wrath of the Lords of the West.

I don't think we know who 'some' are here, but if we work with Tokien's idea that he found and translated a copy of the red book, it presumably means some Elvish writers (whose work Bilbo read in Rivendell). As to what they thought, the last sentence is rather ambiguous: if Sauron acted only out of fear and a wish to save his own skin, then that isn't true repentance at all.

All we can conclude is that Sauron definitely did express regret for his deeds, but he may in fact have been lying.

As much as it pains me to deal with any mention of the show that shall not be named (similar to Elrond's expression here)...

Somewhat. At the start of "Of the Rings of Power and the Third Age" we find

When Thangorodrim was broken and Morgoth overthrown, Sauron put on his fair form again and did obeisance to Eonwe, the herald of Manwe, and abjured all his evil deeds. And some hold that this was not at first falsely done, but that Sauron in truth repented, if only out of fear, being dismayed by the fall of Morgoth and the great wrath of the Lords of the West.

I don't think we know who 'some' are here, but if we work with Tokien's idea that he found and translated a copy of the red book, it presumably means some Elvish writers (whose work Bilbo read in Rivendell). As to what they thought, the last sentence is rather ambiguous: if Sauron acted only out of fear and a wish to save his own skin, then that isn't true repentance at all.

All we can conclude is that Sauron definitely did express regret for his deeds, but he may in fact have been lying.

In response to a follow-up question in the comments, I'm going to go out on a limb ... aside from the above, I do not think there are any instances in which Sauron expresses any type of regret for his actions, false or otherwise. It's always dangerous to say something doesn't exist in a large body of work, but I'm fairly sure I'm correct in this case. After all, Sauron really isn't a very nice chap.

As much as it pains me to deal with any mention of the show that shall not be named (similar to Elrond's expression here)...

Somewhat. But he may well have been lying. At the start of "Of the Rings of Power and the Third Age" we find

When Thangorodrim was broken and Morgoth overthrown, Sauron put on his fair form again and did obeisance to Eonwe, the herald of Manwe, and abjured all his evil deeds. And some hold that this was not at first falsely done, but that Sauron in truth repented...

I don't think we know who 'some' are here, but if we work with Tokien's idea that he found and translated a copy of the red book, it presumably means some Elvish writers (whose work Bilbo read in Rivendell). As to whether Sauron really did really regret his deeds (however briefly), we can only guess.

As much as it pains me to deal with any mention of the show that shall not be named (similar to Elrond's expression here)...

Somewhat. At the start of "Of the Rings of Power and the Third Age" we find

When Thangorodrim was broken and Morgoth overthrown, Sauron put on his fair form again and did obeisance to Eonwe, the herald of Manwe, and abjured all his evil deeds. And some hold that this was not at first falsely done, but that Sauron in truth repented, if only out of fear, being dismayed by the fall of Morgoth and the great wrath of the Lords of the West.

I don't think we know who 'some' are here, but if we work with Tokien's idea that he found and translated a copy of the red book, it presumably means some Elvish writers (whose work Bilbo read in Rivendell). As to what they thought, the last sentence is rather ambiguous: if Sauron acted only out of fear and a wish to save his own skin, then that isn't true repentance at all.

All we can conclude is that Sauron definitely did express regret for his deeds, but he may in fact have been lying.