Great rivalries come and go. First of all, both teams have to be good. Yeah, we all love the old classics that everyone has mentioned here, but they're only relevant if both teams are good for some amount of time. And they have to have a real hatred for each other. And 1 teams has to be the one that is always slightly underdogged and just can't get ahead.
So in the last few years or so I'd go with:
Pistons/Pacers. Ended in a huge brawl that was the beginning of the end for the Pacers.
I don't even like either team, but those games were more intense than any other. Crazy.
Red Wings/Avalanche was pretty crazy too. Hockey, fight, hockey, fight, fight, hockey.
Baseball doesn't bring it like it used to. Teams don't fight as much. Sox/Yankees was pretty good before the Sox were champions. Those teams hated each other and had a huge brawl.
The NFL is harder because you don't always get to play a lot. I'd probably go with Patriots/Colts but I'm not sure there was a lot of hatred there. Packers/Vikings has been pretty crazy over the last 15 years. Players getting into fights with fans, teams taking others legendary QB's. Moss mooning the fans and then having their bus trashed. Been a lot of craziness between these 2 teams fans.