For me Georgetown is always enemy #1. When I enrolled at Villanova in 1982 Georgetown, with Patrick Ewing, was at or near the height of Hoya Paranoia and the best team in the country for a couple years. Georgetown was always the biggest game of the year and the hardest ticket to get. The fact that I was a student at Villanova in '85 when we beat Georgetown makes any victory over Georgetown that much sweeter and any defeat that much more painful.
Number 2 is probably Marquette and it isn't because I dislike them for any reason, in fact, quite the opposite. I like Marquette and think they have the best uniforms in the Big East and top 3-5 in the country. I have a number of family members who went to Marquette (mother, father, sister, uncle #1, uncle #2, cousin, grandfather, former co-worker) so we always have some trash talk prior to the games along with a token bet for a beer or two. The fact that Villanova has won the majority of the games against Marquette means I haven't had to pay up very often.
Plus we always have something in common to talk about.
After that it is probably St. Johns and Providence. The fact that SJU regularly beat VU when I was a student and my general apathy toward NYC (I think it is waaaaaaaaaaaaaay overrated) puts them next on my list. They were down for so long that my dislike became more ambivalence but with their recent resurgence, it is back to dislike. I think the SJU-VU game could be a real problem for Villanova based on the way they played Creighton played Sunday night. I have no data to back this up and to lazy to look but no matter how good or bad they are, Providence always seems to play VU tough and has beaten VU a number of times when it seemed like they had no business doing so.
I have no real strong feelings one way or the other for the other schools. They are just opponents although after being taken to the woodshed by Creighton a few weeks ago, I would very much like to see Villanova win in Omaha on Sunday although I don't expect it to happen.