Please suggest BYOB restos downtown and North of downtown.
Both Schwa and Bonsoiree are BYO. Both have very creative, innovative menus. Both offer tasting menus (Schwa only has tasting menus -- one 5-course, the other 9-course). Both pretty much require reservations and it is much easier to get into Bonsiree than Schwa (which seats 30; reservations taken 2 months in advance).
Both restaurants are north of downtown. Schwa is on Ashland, just south of North Avenue; Bonsiree is on Armitage, west of Western Avenue.
There are a huge number of wonderful BYO all around the city. For the most part, smaller ethnic or casual places, although Schwa is the big exception as one of our top restaurants. Also, many restaurnants have a $10-25 corkage fee if there is a special wine that you want to drink that is not on their menu. I'd also recommend going to lthfoum.com and checking their great neighborhood restaurants list.
A couple of standouts that come to mind:
TAC Quick Thai
Coalfire Pizza (on Grand)
Glen's Diner (serve wine but had a nominal corkage fee last time I went.)
Mixteco (outstanding Mexican on Ashland)
Kan Zaman (Lebanese)