Continuity mistake: When Travis is with the gun salesman, he is trying out the black gun. You see a close up shot of him holding the gun with his arm outstretched with his jacket on. In the very next shot (no time has elapsed) his arms are at his side and his jacket is now completely off.
Continuity mistake: While De Niro is talking to Jodie Foster you can see a bus out the window. When the screen switches to Jodie Foster's view you can see the same bus driving, but then it changes back to De Niro's view you see the same bus driving up to the window again. This keeps happening throughout the scene, the bus keeps driving around the building.
Continuity mistake: When Robert Deniro shoots the pimp's hand off in the hallway (shortly before shooting Sport the second time), blood splatters on his face, across his left eye and forehead. As he stumbles up the stairs (still shooting the handless pimp), the blood has changed, and finally in the shot where Deniro pantomimes shooting himself, the only blood on his face is on his chin.
Continuity mistake: In the scene where De Niro shoots and kills the man robbing the convenience store, the clerk takes De Niro's gun off of him and starts beating the already dead body with a pole. If you watch the scene carefully, the dead body changes positions contrary to how the man is hitting him. Clearly the scene was shot several times with the dead body being placed in different positions.