You know I dont know if in todays age "Rivals" exist, so many teams so many players being friendly with each other (From a pro persepctive)
If I had to pick for my local teams:
Tigers: Indians, White Sox
Wings: Chicago, Toronto
Lions: Um, Next question...But in the days of 9-7 and 8-8 years I would say bears and packers
Pistons: Cavs, Celtics..a few years ago Pacers/Pistons was a nice rival...
U of M: Ohio State
The VIKINGS are the most hated team in Detroit when it comes to football. I'll add Montreal to the Wings list, although we don't see them much anymore. And there still is a special place in our hearts for the Avalanche. If you ask the Red Wing players right now, they'll tell you there is no love lost for San Jose, either.
As for MICHIGAN - MSU and Notre Dame come right after "that evil school from the South".