The top three I found:
Best Picture: Shakespeare in Love over Saving Private Ryan
Music: Hanz Zimmer not even nominated for Blackhawk Down
Directing: Kevin Kostner for Dances With Wolves beats Scorsese's Good Fellas.
DanDarko
Favorite Answer
Forrest Gump beating The Shawshank Redemption and Pulp Fiction
Gran Torino not getting a single Oscar nomination
annyrutts
Mickey Rourke should have won Best Actor for The Wrestler
Chris Nolan should have been nominated for Best Director this year
Martin Scorsese has been snubbed at least 5 times.
Anonymous
shakespeare in love over saving private ryan.
Forrest gump over Pulp Fiction.
Glory not being nominated for best picture (1989)
Dark Knight not being nominated for best picture (probably why there are ten nominees now.)
a serious man by the coens not being recognized that much in 2010.
Recently, NOLAN NOT BEING NOMINATED!!! ARGH.
Anonymous
Robert Redford's "Ordinary People" over Scorsese's "Raging Bull".
TheGrandOnion
Malcolm X was 'epic' yet got no nomination for Best Picture.