Continuity mistake: In the scene where one of the "high council" is speaking (behind a desk or pedastal), there is a close-up shot of the highest ranking council member's face. In his nostril you can see a dry booger. They then change the shot and then come back to a close-up of his face. The booger is no longer there. Sorry if this is a bit gross, but I found it amusing.
Continuity mistake: When Rico is activating the ABC-robot, he tells it "we're going to war". His face is then very close to the robot, only inches away. The angle changes to a shot over Rico's shoulder, and his head is now much further away from the robot, even before the robot starts to sit up.
Continuity mistake: When Dredd and Fergie are fleeing, as the approach the dead end, you see the bar sign in front of them, on the right, which they start to pass when the taxi crashes. In the next shot, they pass it again, but in the next shot after that, the bar sign is once again in front of them.
Continuity mistake: The final battle inside Liberty's head - Rico tries to escape on his Lawmaster motorcycle. Dredd pulls Rico off and the Lawmaster plummets to the ground. Seconds later when they show a slow motion shot of Rico falling to the ground there is no wreckage whatsoever of the Lawmaster beneath him.