Continuity mistake: When Spartan and Huxley arrive at the cryogenics place to battle Phoenix and what remains of his gang, Phoenix sees them on the monitor walking into the building and you see Spartan on the left and Huxley on the right. The camera angle changes to an inside view and Spartan is now on the right.
Continuity mistake: When Spartan and Phoenix are fighting in the sewers, Edgar Friendly shoots his gun at Phoenix and you see the catwalk that Phoenix was on tip to its side and Phoenix dropping down to the ground with the plank of the catwalk right below him. When Phoenix lands on the ground the plank is nowhere to be seen. It should have been right below him.
Continuity mistake: When John Spartan is frozen in the cryochamber, at the beginning of the movie, if you watch during the credits - while the time is going by the small circular monitor that is on his chest, it changes from the left side to the right side of his chest in almost every shot.
Other mistake: During the scene where Phoenix is questioning the leader of the future society in his office, he says at least one word (hell) that he should have been fined for, and isn't. We see that there is a ticket printer in the room in a later scene, when Savage sets it off during his own confrontation.