Two of the more troublesome modal auxiliaries are may and might. When used in the context of granting or seeking permission, might is the past tense of may. Might is considerably more tentative than may.
May I leave class early?
If I've finished all my work and I'm really quiet, might I leave early?
In the context of expressing possibility, may and might are interchangeable present and future forms and might + have + past participle is the past form:
She might be my advisor next semester.
She may be my advisor next semester.
She might have advised me not to take biology.
Avoid confusing the sense of possibility in may with the implication of might, that a hypothetical situation has not in fact occurred. For instance, let's say there's been a helicopter crash at the airport. In his initial report, before all the facts are gathered, a newscaster could say that the pilot "may have been injured." After we discover that the pilot is in fact all right, the newscaster can now say that the pilot "might have been injured" because it is a hypothetical situation that has not occurred. Another example: a body had been identified after much work by a detective. It was reported that "without this painstaking work, the body may have remained unidentified." Since the body was, in fact, identified, might is clearly called for.
Can versus May :
Whether the auxiliary verb can can be used to express permission or not — "Can I leave the room now?" ["I don't know if you can, but you may."] — depends on the level of formality of your text or situation. As Theodore Bernstein puts it in The Careful Writer, "a writer who is attentive to the proprieties will preserve the traditional distinction: can for ability or power to do something, may for permission to do it.
The question is at what level can you safely ignore the "proprieties." Merriam-Webster's Dictionary, tenth edition, says the battle is over and can can be used in virtually any situation to express or ask for permission. Most authorities, however, recommend a stricter adherence to the distinction, at least in formal situations.