NOT A TEACHER
Hello, Tufguy:
1. Yes, it should be "I am a fan of Madonna" in formal writing, and it should probably be "I am a fan of Madonna" in 99% of cases in informal speech and writing, too.
2. When you get some extra time, google "a fan of" -- with the quotation marks.
3. I have just done that, and you will see that the possessive is, however, very idiomatic (the way native speakers use their language).
a. Here is something from a grammar helpline called englishforums.com.: "I am a fan of Sting's."
b. According to one comment, such a sentence "implies (or nearly so) that Sting personally knows you."
4. Now is here a person's comment in a discussion forum called topix.com.: "I am a fan of Beyoncé's but I haven't been impressed by her lately."
a. Of course, I do not know the background of that poster, but I am making an educated (?) guess that the poster and Beyoncé are not personally acquainted. So probably the poster should have avoided the possessive.
5. It seems that using the possessive with a noun is very idiomatic, but probably we should avoid it -- unless we have a close relationship with that person: "My wife's father is Tom Smith. I'm an admirer of Tom's, for he is such a wise person."
6. On the other hand, we have NO choice when it comes to pronouns: "I'm an admirer of his."