One thing I've taken from everyone's responses is that there are a number of factors that determine a rivalry. Animosity between the teams, series history, how evenly-matched the teams are, same conference. Here's a breakdown on some of the contenders. All time records in parenthesis
Cinci (15-24), L'Ville (23-34)
These 2 can be grouped together becuase they are similar. We've played them a number of times, have losing (but competitive) records against both, still play in their conference, and their games definately have the feel of a rivarly game.
Wisconsin (51-61)
Another losing, but competitive record. The best-in-state title adds a lot of fuel to this rivalry, plus the fact that the outcome plays a hand in in-state recruiting battles. Definately feels like a rivalry.
Notre Dame (32-75)
Non-competitive losing record, although we've had the better of them recently. Games have a good rivalry atmosphere and there's definate animosity on our end. I loved the "thumbs up" MU on front, "middle finger" ND on back t-shirts. Unsure,though, how ND students feel about this rivalry.
Schools we've played a lot and for a long time.
DePaul (61-42), Detroit (60-29), Creighton (48-28), Loyola (46-35), SLU (31-23)
Although we have a storied history with these schools, the games don't have the same rivalry feel as some of the others mentioned. DePaul has a chance to become huge again, especially given that they should have a really good team this year.
Pitt (3-3) G-Town (2-2)
Too young, but definately picking up speed. If all our games are as competitive as last year, these could be topping the chart pretty soon. Plus G-Town and MU (and Gonzaga) are competiting to be the best Jesuit basketball school.
UWM (34-0)
Sorry, but a rivarly needs to be competitive. Yes most of the 34 wins were with UWM as a DII school, but there's still 34 wins. If UWM takes 2 of 5, though, watch out. This could be the type of game that leads to rioting.