Question

1. May I come in?
2. Yes, you may.
3. No, you mayn't.

(Can we say #3?)

4. The rumor may not be true.
5. The rumor mayn't be true.
(Can we use the contracted form, 'mayn't'?)

Answer 1

The contraction "mayn't" is a legitimate word, but it's not usually used. I've never heard anyone say this, not even college professors!!

The uncontracted version is what is in normal speech.

No, you may not.
The rumor may not be true.