E. We use ‘may‘ or ‘might‘ to say that something is a possibility. (e.g. It may be true./ It might be true (= perhaps it is true))
a. use ‘might‘: When we are not sure about something. (e.g. I might see you tomorrow.)
b. can use ‘might’ as the past tense of ‘may’ for requests. (e.g. He asked if he might borrow the car.)
c. can use ‘might‘ for very polite requests. (e.g. Might I ask you a question?)

E1. For the past we use ‘may have (done)‘ or ‘might have (done)‘ (e.g. A: I wonder why Kate didn’t answer the phone. B: She may have been asleep. (= perhaps she was asleep))

E2. ‘Could’ is similar to ‘may and might’. (e.g. It’s a strange story, but it could be true. (= it may/ might be true))
a. But ‘couldn’t’ (negative) is different from ‘may not and might not’. (e.g. Sarah couldn’t have got my message. Otherwise she would have replied. (= It is not possible that she got my message))

E3. use ‘may’ and ‘might’ to talk about possible actions or happenings in the future. (e.g. I haven’t decided yet where to go for my holidays. I may go to Ireland. (= perhaps I will go there))
a. Compare: I’ll be late this evening. (for sure) vs. I may/ might be late this evening. (possible)
b. use only ‘might’ (not may) when the situation is not real. (e.g. If I were in Tom’s position, I might look for another job.)

E4. also a continuous form: may/ might be ~ing. Compare: Don’t phone at 8.30. I’ll be watching the film on television. vs. Don’t phone at 8.30. I might be watching (or I may be watching) the film on television. (= perhaps I’ll be watching it)
a. also use ‘may/ might be ~ing’ for possible plans. (e.g. I’m going to Ireland in July. (for sure) vs. I may be going (or I might be going) to Ireland in July. (possible))
b. But you can only say I may go (or I might go) ~ with little difference in meaning.

E5. We ‘might as well’ do something = We should do it because there is no better alternative. There is no reason not to do it. May as well is also possible. (e.g. A: What time are you going out? B: Well, I’m ready, so I might as well go now.)

E6. We make questions by putting the subject after ‘may/might’. (e.g. May I ~ ? Could I ~ ? Might I ~ ? etc.)
a. The negative forms are ‘may not’ and ‘might not’ (or mightn’t). (e.g. It may not be true. (= perhaps it isn’t true))
b. to make polite requests: May I borrow the car tomorrow?
c. When we use ‘may not‘ for a refusal it is emphatic: You may not!