can. (3) permission: In informal circumstances, since the second half of the 19c., can has often been used in contexts of permission where may had earlier been obligatory: Can I speak with the Count? – Tennyson… But in any context where politeness or formality are overriding considerations, may is the better word. May I come and stay with you? —Fowler's Modern English Usage (10-10)
345. 1. permission: can / could more common Can and could are more common thanmay and might, which are used in a formal style. Compare: Can I look at your paper. Excuse me, may I look at your newspaper for a moment? There is an old belief that may/might are more 'correct' than can/could in this case, but this does not reflect normal usage.—Swan(345.1)
can, may 1. The use of can in a direct question to request permission is basically an oral use. (Several examples are given for speech.)
Conclusion: The uses of can which request permission are seldom found in edited prose. In general, this use of can belongs in speech, reported or fictional. In negative statements,cannot and can't are much more frequently used than may not and mayn't; use in negative contexts is seldom notice or criticized. —Merriam-Webster's Dictionary of English Usage(218)
9-2 POLITE REQUESTS WITH "I" AS THE SUBJECT May and could I are used to request permission. They are equally polite. Can I is used informally to request permission, especially if the speaker is talking to someone s/he knows fairly well. Can I is usually considered a little less polite than may I or could I. —Azar (152)
FILL IN THE BLANKS WITH MAY OR CAN:
1) ________ I go to the bathroom?
2) _________ I use my cellphone?
3) _________ I borrow your pencil?
4) _________ I go to the party?
5) _________ Call you tomorrow?
6) _________ I have some more ice-cream?
7) _________ I speak to your sister?