Quick answer
Мо́жно войти́?
mózhna vaytí — or just Мо́жно? — a complete question on its own
The options
Мо́жно войти́?May I come in?
Мо́жно?May I? — complete on its own
Разреши́те?May I? — formal, also for squeezing past
Я не помеша́ю?Am I interrupting?
Вы за́няты?Are you busy?
Мо́жно on its own
At a door, a lift or a table, a single Мо́жно? with a gesture is polite and complete. It covers “may I come in”, “may I sit here”, “may I take this” — the situation supplies the rest.
Разреши́те
More formal, and the word Russians use to squeeze past someone in a crowded metro car: Разреши́те! — allow me. In an office it is the respectful version of Мо́жно войти́?, and Разреши́те войти́ is near-military in tone.
What you hear back
Да, коне́чно.Yes, of course.
Входи́те!Come in!
Одну́ мину́ту, пожа́луйста.One moment, please.
Я за́нят, зайди́те по́зже.I’m busy, come back later.
Typical mistakes
Мо́жно я войду́?
Мо́жно войти́?
мо́жно takes an infinitive. The version with я exists but sounds childlike.
Мо́жно входи́ть?
Мо́жно войти́?
A single act takes the perfective.
Check yourself
What follows мо́жно?
Мо́жно войти́ — infinitive.