"Pa" Angel introduces his elder son (played by Phil Smeeton) as Link, but in the closing credits the character name is Fink. According to comics, Link, Junior and Mean Machine have a brother named Fink, but he does not appear in the film.
When Dredd and Fergie are escaping from security after Rico and Griffin murder the high council, Dredd is wearing full Judge uniform. When he enters the room with the hoverbike, he is wearing the shirt. When he sits on the bike, he has lost the shirt and is wearing only a t-shirt. This happens in timespan of him sitting on the bike, he did not have time to remove it.
All the Supreme Council Judges wear shoulder rank devices. Chief Justice Griffin's (Jürgen Prochnow) are on backwards most of the time. A notable scene is in which the Janus files are unlocked; his shoulder devices flip flop depending on the angle.
When Hershey asks Central to analyse an old photo of Dredd with Rico at the academy, Central gives the description "Unknown male, approximately 200 centimetres tall". 200 cm is nearly 6 feet 7 inches, Armand Assante (who plays Rico) is nowhere near this unusually tall height, his height is less than 6 foot.
Before the enter the exhaust hole, the fire is being blown outwards.. When Dredd and Ferguson run in the exhaust pipe, the flame is chasing them rather than moving away them. This is because Ferguson fell and had no chance of reaching the interior end of the tube, so Dredd and Ferguson began running back out to the exterior end, ahead of the flames going out.
The goof item below may give away important plot points.
The secret feature of the Lawgiver II sidearm, that it registers the user's DNA and imprints it on each bullet fired, is explicitly stated to key on contact with the user's palm - however, we're shown several times that the judges wear heavy gloves while on duty (the first shots of Judge Dredd include one of his gloved hand on the Lawmaster throttle).