1. She enjoys Mary's singing.

Question

2. She enjoys Mary singing.
3. She enjoys his singing.
4. She enjoys him singing.
[Which ones are correct?]

Answer 1

1 and 3 are correct.

Answer 2

extra credit: why?
I assume you are studying gerunds ...

Answer 3

Must be!

http://grammar.ccc.commnet.edu/grammar/gerunds.htm#possessive