Continuity mistake: When the three ladies of the Soul Food Chorus get up to sing "Think" with Aretha Frankin, their food is sitting on the counter; ten seconds later in a wide shot, it has gone. (This is undoubtedly so that "Blue" Lou can do his thing on the counter.) When the ladies sit down again and get a nasty look from Aretha after Matt and Lou have left, their food is back again.
Revealing mistake: In the beginning scene when they jump over the drawbridge there is more than a foot of space at the bottom of the rising bridge. No car, no matter how big, could have cleared that space without flipping over. Also when they land on the other side you can see a "flap" at the bottom of the bridge which would allow the car to drive over it.
Continuity mistake: When the Blues Brothers go to persuade Matt and "Blue" Lou to rejoin the band, and they have just ordered their signature meals from Aretha Franklin, she goes to Matt in the kitchen to moan. Matt is cooking two burgers of some kind, but in a shot of Aretha 30 seconds or so later (when Matt says not to talk that way around his old lady), the burgers have moved and multiplied.
Continuity mistake: When Carrie Fisher is shooting at Jake and Elwood in the tunnel under the Palace Hotel, she hits a water pipe and water sprays everywhere in a mist. In the next shot of Jake and Elwood lying in the mud, there's no spray visible behind them at all. When Jake stands up a few seconds later, the spray is back, but is more directed now.