Here is my understanding of it, from the point of view of a native speaker of AmE.

The pronomialized versions require the possessive pronoun:  of mine, of yours, of his, of hers, of ours, of theirs.
The nominal versions can take either form:  of Mike, of Mike's; of Ann, of Ann's

[a friend of, an acquaintance, a buddy of, a chum of, a pal of
a classmate of, a schoolmate of, a teammate of, a colleague of, an associate of
a fan of, an admirer of
an enemy of, an opponent of, an adversary of, a competitor of]

[mine, yours, his, hers, ours, theirs, Mike, Mike's, Ann, Ann's]
_______

In the case of the nouns, a subtle distinction can be made between the two possibilities.

The form with 's is more intimate in shading.
A friend of Mike's means one of Mike's friends, one selected from among all of Mike's friends.  The idea is that these friends belong, in some sense, to Mike -- that Mike has these friends.  Mike counts this "friend of Mike's" among his friends.

The form without the 's is cooler -- more remote -- in the impression it gives.
A friend of Mike means a person who has friendly feelings toward Mike, but it involves Mike much less.  Mike need not even know that this person has friendly feelings toward him.  This is why we would say something like This organization is a friend of the poor -- never of the poor's!  The poor as a group could never 'personally' choose friends.
___________

Of the several relationships shown above which allow the distinction, there are some which, by their meaning, i.e., not by rules of grammar, would cause us to select the version without the 's more often than not.

For example, one can be an admirer or fan of someone without that person even knowing that you are an admirer or fan, so the version without 's is going to be seen more frequently, thus: I am a fan of Sting.  To use the alternative is much too intimate:  I am a fan of Sting's, because it implies (or nearly so) that Sting personally knows you and counts you personally among his fans.  For the same reason -- in the case of dead artists -- it would be very strange indeed to say such things as I am an admirer of Mozart's!

Unfortunately, the grammar of English does not allow the distinction when the pronomial forms are used:  Charles was an admirer of theirs -- never of them.  A complete rephrasing would be needed if the distinction were important enough to be highlighted.

CJ