I have never found myself "rooting" for Michigan. That said, and of course this is in games other than the Ohio State game, I don't "root" against them. Well, I do root for Minnesota against them, but that's just cause the Gophers are like... my little brother or something. And, once I turned on the Appy State game in the waning moments, I did indeed actively root for Appy State to beat Mich. But, had that game been against... Nebraska, I would not have done so.
In any event, if I root for anything in a Michigan game, I root for sloppy play and mental mistakes. When Michigan wins, I don't care, it's fine (Even against Minnesota, not against OSU of course). When they lose, I find myself enjoying it, even if ever so briefly. But, I can't say I root for them to lose.
My favorite ND v. Mich game is when both teams trip over themselves trying to get the other team to win. I like goal line fumbles. I like poorly thrown picks. I like bad coaching. I forget the precise year - it was 1996 or close to it - but it was the perfect ND v. Mich game. Neither team wanted to win, and they played like it. Fumbles, bad play calling, drops, INTs.... poor clock management.... It was great.