Continuity mistake: In the scene after Queenan dies, when Sullivan picks up his phone to contact Costigan, he presses the "talk" button and a smudge of blood is left. In subsequent shots, there's no trace of blood on the "talk" button.
Other mistake: When Costigan drags a handcuffed Sullivan into the elevator, Costigan slams Sullivan against the wall and his left arm is visible and not in handcuffs.
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 Costigan is with Madolyn in her office, the tissues in the tissue box change consistently between shots.
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: 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: When Costello is hitting Costigan's broken wrist, as Costello is leaving the room, the shot shows Costigan crumpling to the floor beside the pool table. He is on the floor as Costello is leaving. But, the next shot shows him back on the pool table crumpling to the floor again from a bird's eye view.