Rosa's Lounge on West Armitage.  

http://rosaslounge.com/index.cfm

The top places IMO (in no particular order).  Of course it all depends on who's playing where on a particular night.

Kingston Mines
Buddy Guy's Legends
B.L.U.E.S.
Checkered Board Lounge
Blue Chicago
Rosa's

http://www.chicagobluesguide.com/guide/blues-clubs/blues-clubs-page.html
http://centerstage.net/lifestyle/articles/askcenterstage_bluesclub.html

a few blocks from second city off lincoln on a dead end street.  If coming from Lincoln n Armitage, go south on Lincoln to the first street on left and go north on that street about a half block and it dead ends at a circle and on the left side corner there used to be a place just below street level I believe.  Haven't been in over 20 years but for my money, they were one of the best in Chicago.  

Maybe a current Blues club runner can come up with the name and current status of the place.

I had a student whose husband was Buddy Gye's manager. They gave me tickets and a comp for a meal. Loved the jambalaya.

www.blues-bar.com/  This is a nice blues bar

2 West Busse Avenue  Mount Prospect, IL 60056
(847) 788-9977