(poll to come)
For me, it would be:
[ul]
[li]Barcelona[/li][li]Rangers[/li][li]Red Sox[/li][li]Bears[/li][li]Cowboys[/li][li]Lakers[/li][li]Pakistan[/li][li]North Carolina[/li][li]Arsenal[/li][li]Notre Dame[/li][li]Auburn[/li][li]Texas[/li][li]Argentina[/li][li]Germany[/li][li]Boca Juniors[/li][li]Maple Leafs[/li][li]Ohio State[/li][/ul]
(You don’t have to list the whole list; any or a few of your preferences are fine)
Not that I support the Red Skins but I support the team playing the Cowboys.
BTW: The NFC East is all rivals. Boys vs Skins is hardly the biggest.
Ask me again sometime when I have my flipping coin handy.
No Cubs vs. White Sox?
Or Cubs v. Cardinals?
Or Dodgers v. Giants?
Or, for that matter, where is University of Washington v. Washington State University?
What about Lehigh and Lafayette? The longest running rivalry in college football!
Celtics over Lakers
Schalke over Bayern Munich
Puma over Addidas
Converse over Nike
Mackoy over Hatfield
Ohio State (vs *ichigan)
Cleveland Browns (vs Pittsburgh)
North Carolina (vs Duke)
Georgia (vs Georgia Tech)
There are well over 100 rivalries that could have been listed in the poll. Some deserving ones were going to have to be left off. The poll is already 34 voting options long.
I support Real Madrid because I have a co-worker who supports Barcelona and I love getting him all worked up.
I’m an Ohio Sate fan because I grew up among others of the same persuasion.
I support the Germany because I’m American so I need a backup national soccer team and it was either them or Sweden.
I’m a Red Sox fan because anything else is too horrible to contemplate. 
With the single exception of Bears versus Packers (I’m from Chicago), in all of the rivalries listed for sports that I follow (baseball, football, and basketball), I hate both of the listed teams.
my mom who grew up in Michigan for some reason unknown to everyone was a life long fan of the Dodgers in Brooklyn and LA to the point that if you were a giants fan you hoped she never found out as she wouldn’t acknowledge you existed …
I’m from Pittsburgh so I pretty much support all of our teams over any of our rivals. (Most of them are from Cleveland or Philadelphia)
NFC East is by far the best division in NFL football. NFC Central a solid 2nd though.
Dodgers v Giants for absolutely sure.
I’m a Cardinals fan. The Cubs-Cardinals another good rivalry, but having lived in California, the Dodgers-Giants rivalry is a lot more intense. It’s all about Nor-Cal vs So-Cal.
I hate Duke no matter who the Blue Devils are playing.
To paraphrase Sir Winston, if Duke were playing the spawn of Hell, I would contrive to say something good about Satan.
Also whoever is playing Notre Dame, Ohio State, the Chicago Cubs, the N.Y. Yankees…
I support every one of these, except the first - I can’t stand Ohio State.
I also voted for the Packers over the Bears. And I’d add Georgia over Florida, Georgia over Auburn, and North Carolina over Virginia (the South’s oldest football rivalry). And I’ll cheer for any team playing the Ravens, Cowboys, or Patriots.
I don’t follow the sport, so I don’t have any preference in any of the soccer rivalries you listed.
Surprised you didn’t list Harvard-Yale, though - that’s pretty famous.
It used to be I could root for a tie in the Alabama-Auburn game so that neither one would have a win. Stupid overtime rule.
I’m not for any of the teams, but will gladly vote against Rangers, Yankees, Cowboys, Ohio.