Revealing mistake: When Costello's men drive off, after throwing Queenan off the top of the building, the state troopers are firing at the van. One back window is shot out, then in the next shot of the van the other window is shot out and then the same one is shot again so the one window is shot out twice in the same scene.
Continuity mistake: When Costello and French encounter the cops, the first shot shows the cops' car turning their lights on. In a new shot a second afterwards the lights on the middle and right side cars are out and they are turning them on again. On yet another new angle, the lights on the car on the left turn from off and on again in a split second.
Continuity mistake: In the scene after Queenan dies, when Sullivan picks up his phone, the number Sullivan calls Costigan in isn't the same as the one from which Costigan calls back, even though it's the same mobile (silver and blue vs. Costigan's red phone for Costello's gang). When Sullivan calls, the number begins with something like "617-854" and when he sees Costigan calling back, it's "617-177".
Continuity mistake: While Queenan is talking to Costello out on the street, watch the people behind him. First there's no one, then we have a shot of a woman and a huge line of kids passing next to her. Then back to Costello and the woman, who hasn't moved, is now miraculously visible behind him. To add to the mess, the kids have vanished.