It's generally more polite in English to say something about your own feelings, like "I'd prefer you didn't..." or "I'm afraid I can't..." And if you want to be even more polite, phrase it as a question or suggestion, or add a softening word like "I'm afraid..." or "Sorry, but...". For example: "May I open the window?" "Actually, I'm a little cold - would you mind leaving it shut?" "May I sit here?" "Sorry, but I'm waiting for someone." "May I borrow your pen?" "I have a test soon, would you mind asking someone else?"

This is a fairly difficult question, I think. "No you can't!" is definitely the rudest of the three options, I think. But depending on tone, "No you may not! and "No you mustn't!" could both be valid answers. I think I would select "No, you mustn't!" because it would almsot be received like a plea. With any tone implied by the use of an exclamation point (!) "No you may not!" is probably going to be rude. "No you mustn't!" sounds plaintive and beseeching. I would select that from the listed options.

Devoid of body language, tone etc, "May" is the most polite form. It implies permission. The rest don't. However, in real life, body language and the tone counts even more. Even when using "may", it can be construed as a sarcasm! Smiling sweetly with a box of chocolates in hand and sparkles in your eyes, you asked, "Can I give you a kiss?". This will not be construed as rude or forward. It may be most welcome UNLESS you don't like that person in the first place! The oral speech is only a tiny part in the scheme of effective communication and human nature counts the most!

If you want to be rude, be more creative, but this may be seamed like a challenge the second part.

WOW!... I'm UTTERLY impressed by your perfect answer (without any exaggeration!). In fact when I was publishing this question, I had very good arguments to consider 'No, you may not' the rudest answer. Although, when I received comments on this, I thought that I might be wrong, indeed. Without any doubts, your answer is best! Would you like to ponder some more over becoming an Italki teacher with such astounding English skills?... :-) THANK YOU VERY MUCH!