grammar

I (an American) would normally say “might have done” for a counterfactual or hypothetical:

“I might have done what you wanted, if you had told me what that was.”

“I might have done something very different, if I had known then what I know now.”

In these cases, I’d call “might have” and “if _ had” more of a past subjunctive than a past perfect. It’s analogous to “if I were” instead of “if I was” for counterfactual or hypothetical statements in formal English.

I would normally use may have instead of might have for past-tense statements of fact that are unknown or uncertain:

“I don’t know why the butler would have done it, but he may have.”

“The butler may have done it, but why would he have?”

This is just a past tense of may do, analogous to would he have done being the past-tense form of would do. (We drop “done it” in the second clause of both examples. The listener will understand that the clauses are parallel and have the same main verb.)

The claim that might is the past-tense form of may does not sound correct to me, at least in American English. But also, the distinction between may and might is not strictly observed today, and you will often hear them used interchangeably.

If you phrased a factual event as a hypothetical, it would communicate the opposite: that you’re implying something did happen.

“I can’t say who told him, but I might have spoken to him about it.”

This comes across as ironic understatement. I would understand this to mean that the speaker is not really uncertain about whether they remember speaking to the other person and what they said. This is an unofficial, off-the-record confirmation. I’m being told a secret, with just enough ambiguity that I can say I don’t know for sure.

In that sentence, may would be less clear:

“All these years later, I can’t say for sure, but I may have spoken to him about it.”

This reads to me as if the speaker genuinely doesn’t trust their own memory.