It's hard to choose a favorite "Sean Avery is a dirtbag" moment. For me it came when he decided during the Stanley Cup Playoffs in 2008 to further his rep by seeing how much he could torment Marty Brodeur and get away with.
Much to my chagrin at the time, Avery posted himself in front of Marty with his back to the play and proceeded to stick his stick in the face of Brodeur and just wave it around. Not against the rules, but about as amateur a play as you can find at this level. Even his own teammates didn't know how to react, with some of them yelling at him to get his "stick down" and "knock it off."
But Avery wouldn't stop there, calling Brodeur "fatso" during a post game interview. Makes you wonder why no team seems to keep this guy around for more than a year.