Revealing mistake: In one of the final scenes where Stallone is hanging out of the Statue of Liberty, you can see that his "JUSTICE TATTOO" on his upper arm is a fake. You can see at the edge of the tattoo that it is glued on the skin.
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.
Visible crew/equipment: When asking the street gang in the street war sequence to drop their weapons ("I am the law") you can clearly see the cable the microphone is connected to running from the back of Stallone's helmet. This cable is not visible before or after.
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.
Audio problem: Just before Junior Angel shoot a missile on the prisoner's ship, he says something like "Gonna get me some city boys", but his lips don't move.
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.
Revealing mistake: When Fergie is given his work detail and address at the start of the film he walks through the door and sees Mega City. Look at the live traffic at the bottom of the screen - the same shot is used again just as he boards the yellow shuttle. The traffic queue is the same vehicles in the same order.
Other mistake: As the transport rises and passes close to the lone guard, it's reflected in his visor. However, since it's close enough that the thrusters are billowing his cloak, the reflection should be bigger.