Yankees - Red Sox (Baseball)
Bears - Packers (Football)
Cubs - Cardinals (Baseball)
Broncos - Raiders (Football)
Dodgers - Giants (Baseball)
Michigan - Ohio State (Football)
Duke - North Carolina (Basketball)
Florida - Florida State (Football)
Georgia - Georgia Tech (Basketball, Football)
Others???
Army - Navy (football)
Football –
Cowboys – Redskins
Cowboys – Eagles
Cowboys – Giants
Heh…you sorta took my idea ahead of me:
Giants – Eagles
Giants – Cowboys
Giants – Redskins
And for bonus points, Giants – Jets
Au contraire. It’s Vikings - Packers 
ALABAMA v. auburn
all sports… life… everything
How could I have missed: :smack:
Texas - Oklahoma (All sports and life in general)
Hockey:
I love the Devils. I also love anyone who is playing against the Flyers, Rangers, or Islanders. However, if the Islanders or Flyers are playing the Rangers, I root for them to beat the Rangers. Some Venn-diagrammable permutation of this exists among every fan swearing allegiance to any of these teams.
A smart man roots in this order (his own team excepted, of course):
Umm … and the cup belongs to? …
Chiefs vs. Raiders from the NFL. And of course, we can’t forget to mention the legendary NHL rivalry, the Columbus Blue Jackets against the Nashville Predators. Man, that one has some history.
Arsenal - Tottenham
Everton vs. Liverpool (England)
Boca vs. River (Argentina)
Celtic vs. Rangers (Scotland)
Barcelona vs. Real Madrid (Spain)
I’m sure there are some good national rivalries in soccer as well, but i’m not well-versed enough to know what the best one might be.
In rugby union, i think Australia vs. New Zealand is the big one, although i’m sure some British Dopers would argue with that. England vs. France? Ireland vs. England? 
In rugby league, there’s nothing to compare with the NSW vs. Queensland “State of Origin” clashes.
Cricket is a tougher one. England vs. Australia has plenty of history, but geo-politics probably makes India vs. Pakistan the most interesting right now.
I think it’s worth distinguishing between different types of rivalry. Some rivalries are have extremely bitter origins and can cause lots of real anger (and even violence) among fans. I guess Celtic vs. Rangers might be an example.
Others are rivalries of respect and toughness, and are more concerned with tradition than actual animosity. I think Australia vs. New Zealand in rugby might be a good example of that. I know that, as an Aussie, i love to see the Wallabies beat the All-Blacks, but there’s no hatred or anything. The Springboks, on the other hand…
Good ones all. I add:
Miami-FSU (college football)
Harvard-Yale (everything)
The local ones may be givens, but what the hell…
Giants/A’s
49ers/Raiders
49ers/Cowboys
And i forgot Aussie Rules:
Carlton vs. Collingwood
At least for me, national footy rivalries are
England - Germany
England - Argentina (arrgghhhhh!!!)
USC beating notre dame
USC beating ucla
Ferrari vs. everyone else
Rossi vs. Biaggi (a classic!)
Honda vs. Yamaha
Player vs. The House :rolleyes:
Good vs. Evil (I always bet on Evil)
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Evil will always triumph over good, because good…is dumb.
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Second vote for Celtic - Rangers. I can’t think of any rivalry that could match it.
Toronto Maple Leafs vs Montreal Canadiens!