“May I use your phone?” (Heym)
At any rate she murmured something to the effect that he might stay if he wished. (Hardy)

As has been mentioned earlier, can also often expresses permission but of a different kind:

You may take the book means that the speaker gives the permission;
You can take the book means that there are no conditions that prevent the person from taking the book.

May expressing permission is used only with the Indefinite Infinitive. Here we must observe the difference in the expression of permission and prohibition.
Whereas the former is always expressed by may, the latter has other ways of expression besides may not. Very often the negative answer to the question containing a request for permission is don’t or must not.

May I read the letter? No, don’t, please.

Don’t is less strict than may not, it is rather asking somebody not to do something than actually prohibiting something, which is expressed by may not.
Must not means that it is not the person who prohibits the action, but that there are facts, rules, or circumstances prohibiting it.

You must not smoke so much.

Uncertainty as to the fulfilment of an action, state or occurrence, supposition implying doubt.

“You may think you’re very old,” he said, “but you strike me as extremely young.” (Galsworthy)
She was and remains a riddle to me. She may and she may not prove to be a riddle to you. (Dreiser)

Not which follows may expressing uncertainty is always strongly stressed.

Sometimes when Mr. de Winter is away, and you feel lonely, you might like to come up to these rooms and sit here. (Du Maurlier)

The last example shows that might denoting uncertainty has no temporal meaning, i. e. it does not express uncertainty referring to the past. Practically speaking there is hardly any difference between may and might in this meaning.

Sometimes might expresses greater reserve or uncertainty on the part of the speaker than may.
Might instead of may is often used because of the sequence of tenses.

That was like her — she had no foresight. Still — who knew? — she might be right. (Galsworthy)
Mrs. Page reflected that though Andrew looked hungry he might not be hard to feed. (Cronin)

May denoting uncertainty is used with all the forms of the infinitive.

A Forsyte might perhaps still be living in that house, to guard it jealously. (Galsworthy)

It was then that his voice grew tired and his speech impeded. The knowledge that he had entirely lost touch with his audience may have been the cause. (Greene)

How long the silence lasted the Gadfly could not tell; it might have been an instant, or an eternity. (Voynich)

For all I knew, she may have been waiting for hours.

When uncertainty is expressed the time of the action is indicated by the form of the infinitive and not by the form of the modal verb, as both may and might can refer to the present or to the past in accordance with the form of the infinitive. If the action refers to the past, the Perfect Infinitive is used.

May is used in this sense only in affirmative sentences. Can is also possible in this meaning.

In this museum you may see some interesting things.
You may see him every morning walking with his dog.

In this meaning it is generally used with the Indefinite Infinitive.

Only might is used in this meaning but not may.

You might lend me a razor. I was shaved this morning with a sort of bill- hook. (Galsworthy)

When might is used with the Indefinite Infinitive it is rather a request made in the tone of a reproach, as the above example shows. When it is used with the Perfect Infinitive, it expresses reproach.

I realize now how you spent your days and why you were so forgetful. Tennis lesson, my eye. You might have told me, you know. (Du Maurier)