Modal Verbs: May - Might

May

Used to say that sth is possible: That may or may not be true. He may have (= perhaps he has) missed his train. They may well win. There is a range of programs on the market, which may be described as design aids.
Used when admitting that sth is true before introducing another point, argument, etc.: He may be a good father but he's a terrible husband.
(Formal) used to ask for or give permission: May I come in? You may come in if you wish.
(Formal) used as a polite way of making a comment, asking a question, etc.: You look lovely, if I may say so. May I ask why you took that decision? If I may just add one thing...
(Formal) used to express wishes and hopes: May she rest in peace. Business has been thriving in the past year. Long may it continue to do so?
(Formal) used to say what the purpose of sth is: There is a need for more resources so that all children may have a decent education.
Maybe
Used when you are not certain that sth will happen or that sth is true or is a correct number SYN perhaps: Maybe he'll come, maybe he won't. "Are you going to sell your house? Maybe." It will cost two, maybe three hundred pounds. We go there maybe once or twice a month.
Used when making a suggestion. SYN perhaps: I thought maybe we could go together. Maybe you should tell her.
Used to agree with sb, to add more Information that should be thought about. SYN perhaps: "You should stop work when you have the baby. Maybe, but I can't afford to."
Used when replying to a question or an idea, when you are not sure whether to agree or disagree. SYN perhaps: "I think he should resign. Maybe."

Might

Used as the past tense of May when reporting what sb has said: He said he might come tomorrow.
Used when showing that sth is or was possible: He might get there in time, but I can't be sure. I know Vicky doesn't like the job, but I mightn't find it too bad. The pills might have helped him, if only he'd taken them regularly. He might say that now (= it is true that he does), but he can soon change his mind.
Used to make a polite suggestion: You might try calling the help desk. I thought we might go to the zoo on Saturday.
(BrE) used to ask permission politely: Might I use your phone? If I might just say something...
(Formal) used to ask for Information: How might the plans be improved upon? And who might she be?
Used to show that you are annoyed about sth that sb could do or could have done: I think you might at least offer to help! Honestly, you might have told me!
Used to say that you are not surprised by sth: I might have guessed it was you!
Used to emphasize that an important point has been made: "And where is the money coming from? You might well ask!"