My "home course" is Braemar in Edina (neighboring soutwest suburb of Minneapolis). It is a very nice public course. Play the backside-holes 10-27.
If you want to do a quick executive 9 - Braemar also has that. Additionally, there is a little hidden executive 9 course in Edina that is also owned by Braemer/City of Edina called Fred Richards. An excellent value and typically not very busy.
Dwan in Bloomington (another southwest suburb) is also a nice public course - but is often times very busy.
As mentioned above, the City of Minneapolis has a handful of muni courses. Most are in ok shape and a decent value - but typically pretty busy.
If you want to travel a little bit, I would highly recommend the semi-private course of Mississippi Dunes in Cottage Grove. (about 30 mins south-east of Minneapolis). It is a links style course and sits along side the Mississippi River. Some great scenic holes, nice wild life, very well maintained, and quite challenging.