This is an interesting discussion. It is hard to pick a favorite which is the best because they have all made their share of high quality cinema.
Fincher has made the most masterpieces but what would kind of films would Nolan have done if he hadn't spend most his career on the Batman trilogy? (Not to say that is a bad thing at all)
Requiem for a Dream is also a masterpiece. That film is an emotional roller coaster that stays in your head forever. (The score by Clint Mansell deserves just as much credit as D.A.) That film along with the Wrestler and Black Swan have the best performances out of this whole discussion.
The Fountain is vastly misunderstood. I don't completely understand it myself but I would still give it a better rating than most of the mediocre ratings it has received.
Also, people really shouldn't rate the Girl With the Dragon Tattoo if they haven't read the book. Fincher nailed everything about that book, matching it with his own gritty style.
it would be interesting to see Paul Thomas Andersoson and the Coen Brothers in this discussion. That would make up my top 5 modern directors. Oh, and cannot forget Tarantino either.