Must expresses speaker's obligation, have to comes from outside. Must can be used with adverbs, for focus, or for concession. Must not expresses prohibitions. Don't have to and needn't express lack of obligation
May indicates strong likelihood, might hints at lower likelihood. In casual speech, may and might can be used interchangeably. May and might are modal auxiliary verbs for possibility and permission
Can is used for both asking and giving permission. Could is more formal than can for asking permission. May is the most formal way to ask and give permission
Could expresses possibility in present and future situations. Could is used for making suggestions. Could is more formal than can for asking permission
To go away or not take something with you. To cause someone to stay. To make something available. To give responsibility
May indicates possible or factual situations. Might denotes hypothetical or counterfactual possibilities. Might suggests lower probability than may