Anderson explains to Dredd that she doesn't wear a helmet because it interferes with her psychic abilities. By that logic anyone else wearing a helmet should be protected from Anderson's mind reading ability. However, she is able to easily read Dredd's mind (to the point of almost learning his deepest darkest secret before being told to stop), and immediately determine Judge Kaplan's deception and intent to kill her.
When Ma-Ma asks Dredd if he "like[s] the jewelery" and reveals the transmitter on her left wrist, the shot cuts to Dredd, who turns slightly to look at it. However, he turns to his own left, as if Ma-Ma had the transmitter on her right wrist.
At 1:14:30, Dredd checks down a litany of ammo types, trying to find one that the gun isn't out of. He checks Rapid Fire/RF, Incendiary/IN, & Armor Piercing/AP. Two of these names are visible & have 0 levels showing in the led readout, and the (type obscured) readout is probably Rapid Fire. When he calls for High Explosive/HE, suddenly readouts for all ammo types show 25 rounds each ammo remaining, and the readout for (type obscured)RF, probably, reads 50.
A computer monitor in the Hall of Justice gives one suspect's DOB as being in 2059 and his age as 28, setting the film in 2087. However's the second suspect's DOB is in 2061 and his age as 49, which would set the film in 2110.
When Dredd and Anderson approach a group of 10 thugs outside the Medical Centre, Dredd throws a stun grenade. As the camera tracks the grenades progress across the floor, the wire used to guide it is clearly visible.
While Dredd and Anderson are in Ma Ma's quarters, Anderson gets hit by a bullet that penetrates her vest under her left breast, yet when they are walking out of Peach Trees, Anderson's vest is undamaged.
In the flashback scene showing Ma-Ma's opposition being killed, one man who is obviously alive and conscious is thrown down on the ground and shot several times in the back. As the bullets strike, he doesn't flinch from the impact.
The goof items below may give away important plot points.
When the block goes into lockdown, the blast door comes down and crushes the vagrant right in front of Dredd and Anderson. At the end of the movie, when the blast door goes up, there is no sign of the body.
When Dredd and Anderson first arrive at Peach Trees, they call for some wagons to take away dead bodies and a suspect. No one shows up and control doesn't try to figure out why the judges haven't filed a report yet. Then there is a convenient security system check without any record of authorization right after a call from the judges for the wagons.