I used to consider Detroit our biggest rival. The proximity of the teams to the Michigan-OSU rivalry contributed to this. Also, we got swept out of the first round by the Wings in 09.
Realignment changed that and we no longer share a division with Detroit (whose closest division rivals are now Buffalo and Toronto).
Before last year's playoffs were even over I started to hate the Penguins, and especially their captain Sidney Crosby. In general I don't like any of our division rivals (every other team in the Metro sans Washington has at least one Stanley Cup while we do not), but the Penguins, given the history between Ohio sports teams and those of Pittsburgh (the Browns-Steelers rivalry of the NFL comes to mind), top my list. Second is the Flyers, I can't stand fans of any Philly team, whether they are Flyers, Eagles, Phils, Sixers, Owls, Wildcats, Hawks, Quakers, Explorers, or Dragons (if you can guess what the last 6 are you get a cookie) - from reading stuff online I know how Philly sports fans can be. Third on my list is the Devils - being somewhat (but not fully) religious, the name "Devils" offends me. Also, Rutgers is joining Ohio State in the Big Ten in July.
Because of realignment, I no longer hold any ill will towards Chicago, Nashville or St. Louis either, we now only play those teams twice a year.