Continuity mistake: When Edward Nygma is in his apartment after killing Stickley, we see him pasting letters for the next riddle involving the match head. However, when Bruce and Chase are in her office looking at the riddle after it was delivered, you can see the letters are in different positions than the earlier shot.
Continuity mistake: Bruce's hair is much flatter and messier in the garage than in the previous scene outside when Dick is leaving, even though the garage scene happens directly after.
Other mistake: When Batman puts the lit torch to the cable and the cable breaks; the vault goes back into exactly the same place where the helicopter lifted it out from; connected via a grapple that Batman put through the wall. Ignoring the logistics of how this would be possible, it tightens the grapple, which surely, with the weight of the vault, would have ripped itself away from the wall.
Continuity mistake: As the giant safe is being pulled out of the building, a large section of the wall above the hole breaks and starts falling outside the building. The shot cuts to outside and the hole is now back to the same size as before and no chunks of wall are falling. There aren't even any cracks where a piece might break off.