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How often can you hear an educated native speaker say

She can't have done that, can she? She can have done that, can't she?

She mustn't have done that, must she? She must have done that, mustn't she?

She may not have done that, may she? She may have done that, may she not?

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    I don't think I've ever heard anybody use the question tag "may she not?" – Peter Shor May 6 '16 at 14:53
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    Depends how fast they can speak. ;) – Jonathan S. May 6 '16 at 14:53
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You are asking about a particular way that English frames closed questions (i.e., those with yes or no answers), namely by combining a declarative statement with an interrogative confirmatory tag of opposite polarity. This means that the tag asks the person addressed to agree with or deny the statement by using a negative particle (not, perhaps hidden in the suffix *-n't) when the statement lacks one and vice versa.

So you may make a positive declaration (no not) that

[1a] She is there

and turn it into a question with a negative tag by asking

[1b] She is there, isn't she?

And in reverse, you make a negative declaration (with not)

[2a] She isn't there

and turn it into a question with a positive tag, one without not.

[2b] She isn't there, is she?

tl;dr: Native speakers will tend to avoid phrasing a question in such a manner that the aspect of an auxiliary in a declarative statement does not match the aspect of the auxiliary in the associated tag.

The reason that you don't hear native speakers using some of your phrasings has to do with the aspect carried by English auxiliary verbs like can, could, may, and might. (Aspect refers to the meanings of verbs beyond person, tense, and number conveyed by their inflectional forms.)

Let's start with

[3a] She can have done that, can't she?

Can here means having the ability, and in the present tense it has an enduring aspect, i.e., the ability has been in effect, is in effect now, and likely will continue to be in effect. As in

[3b] She can play the piano.

This conflicts with the past interval of completed action covered by have done. However, we can transpose into the past with could to make the tenses compatible:

[3c] She could have done that, couldn't she?

(Notice the positive could in the statement and the negative coudn't in the tag.)

So why would a native speaker say

[4a] She can't have done that, can she?

In this case, instead of denoting ability, can't with the present perfect carries the aspect of incredulity on the part of the speaker. It means

[4b] I can't believe that she did that, can you?"

Let's examine

[5a] She mustn't have done that, must she?
[5b] She must have done that, mustn't she?

The problem here is that must in the tags has the aspect of compulsion as in

[5c] You must pay your taxes.
[5d] You mustn't exceed the speed limit.

There's an understood "or else". But in the declarative statement in the present perfect carries a different aspect, one of conviction about the outcome of a concluded event. 5a means

[5a-a] She surely didn't do that, did she?

and [5b] means

[5b-a] She surely did do that, didn't she?

The conflict in meanings makes the tag an inapt query for confirmation: the tags ask questions not germane to their associated declarations. Transposition to the present tense keeps the compulsion for both statement and tag, so there's no conflict:

[6a] She must pay her taxes, mustn't she?
[6b] She mustn't abandon her dignity to please you, must she?

The same mechanism is at work for may. Combined with the present perfect in the statement, may has the aspect of doubtfulness. But in the tag, may has the aspect of permission. Thus

[7a] She may not have done that.
[7b] She may have done that.

both mean that it's possible that she did that or it's possible she didn't, but the tags

[7c] May she?
[7d] May she not?

both ask about whether or not she has permission to do something. The clash of meanings precludes 7c from following 7a and 7d from following 7b. We can transpose to the present tense to keep both statement and tag talking about permission:

[7a-a] She may not go to the party, may she?
[7b-b] She may go to the party, may she not?

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