1. She enjoys Mary's singing.

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

1 and 3 are correct.

extra credit: why?

I assume you are studying gerunds ...

Must be!

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

To determine which sentences are grammatically correct, let's examine the verb "enjoys" and how it is used with the gerund form of the verb "singing."

1. "She enjoys Mary's singing." - This sentence is correct because the possessive form "Mary's" is used before the gerund "singing." It indicates that Mary is the one doing the singing, and she possesses the action.

2. "She enjoys Mary singing." - This sentence is also correct. The -ing form of the verb "singing" functions as a gerund here, which acts as a noun. This sentence means that she enjoys listening to Mary's singing without explicitly indicating ownership.

3. "She enjoys his singing." - This sentence is correct. "His" refers to another person, and the gerund "singing" functions as a noun. It states that she enjoys listening to the singing of the person referred to as "his."

4. "She enjoys him singing." - This sentence is not grammatically correct. The pronoun "him" should instead be in the possessive form "his" to match the possessive nature of the gerund "singing." The correct sentence would be "She enjoys his singing."

In summary, sentences 1, 2, and 3 are correct, while sentence 4 requires correction to say "She enjoys his singing" to be grammatically accurate.