Ok, here are my "Must watch", i'll admit they aren't all "classics" in the usual sense but they are all really good.
Romance - Clannad / Clannad - After Story, Koi Kaze, 5 Centimeters per Second
Psychological - Elfen Lied, Death Note, Ima, Soko ni Iru Boku
Mecha - Tengen Toppa Gurren Lagann
Comedy - Sayonara Zetsubou Sensei, Natsu no Arashi!
Action - Gantz / Gantz - Second Stage,
Mystery - Bakemonogatari, Mnemosyne: Mnemosyne no Musume-tachi
Tragedy - Kimi ga Nozomu Eien, Tokyo Magnitude 8.0,
Children's - Cardcaptor Sakura, Pokemon