In some viewers' interpretation, the last segments of the film are a dream sequence, so many "errors" are simply the experience of being in dreams. That interpretation is not universally held, so these goofs are included on this page for your judgment.
The black-market gun dealer misidentifies the .380 Astra Constable in his case as a Walther PPK. The guns are very similar, so it could be an easy mistake or a deliberate lie. He tells Travis the Walther PPK .380 replaced the P-38 as the standard German military sidearm in WW2, which is false. The P-38, a full-size pistol, replaced the P-08 (Luger) as the standard military sidearm. The PPK was a compact pistol for use by police (PP stood for Polizeipistole) and personal protection, not as a combat sidearm.
Easy Andy claims that the .44 Magnum revolver will "stop a car at a hundred yards, put a round right through the engine block." It actually won't penetrate an engine block at that range, and the shooter will have little hope of damaging or disabling a component of the engine that will disable the vehicle (such as shorting out the battery or causing the coolant to drain out). However, it's obviously part of his sales pitch.
According to the exterior shot, and the Then-and-Now Special Feature on the 2-disc DVD, Iris lived at 226 East 13th Street. Travis writes "240 East 13th Street" on the envelope containing the letter and $500.
In an earlier version, Iris's timekeeper discovers a weapon on Travis, disarms him, then returns it to him as he's leaving. The scene was edited out, but the gun is still in the timekeeper's hand when he looks at his watch.
When Travis buys the guns from Easy Andy, he buys a S&W model 61, which he was told was a Colt .25. Later, at the range, he fires a Colt .25. In the second range scene, Travis's Colt .38 Special becomes a Colt Detective's Special.
When Travis is talking to the Wizard, facing the street, flashing lights from a police car appear on Travis' face. The camera immediately goes to Wizard, who is facing Travis, with no police car behind him. Seconds later a police car pulls up, lights flashing, across the street.
When Travis meets with Betsy the first time at the Palantine office, an overhead shot shows him beginning to sweep his arm through the air above the desk. In the next shot, from the front, his arms are in a different position.
When Travis and Iris are eating breakfast, she is spreading jam and sprinkling sugar on a piece of toast. When she places the second piece of toast on her creation to make a sandwich, the two pieces are not aligned as they would be in the loaf. But once they are viewed on her plate they are now aligned.
At the end, (after the police arrive) the camera does an aerial shout over the scene. Sport is still holding his weapon. Whether wounded or dead, the police having passed by, he would not have left the weapon in his hand.
While Travis is examining various guns with the gun dealer, the latter states that the German Walther PPK pistol replaced the P.38 pistol during World War II. In fact, only a small number of PPK's were issued to German forces during the war. Most German sidearms were P.38's and Lugers.
When Travis is buying the guns, he holds the gun in his right hand, but he sights down the weapon with his left eye. Although this is uncommon there are a number of right handed shooters who are "Left-eye dominant". This can be uncomfortable when shooting high powered handguns as the hammer tends to kiss the shooters forehead.
While Travis is checking the .44 Magnum revolver offered by the gun dealer, the cylinder emits a clicking sound while he spins it. In fact, the cylinder of a double action revolver such as this one does not click when it is rotated.
Before the climatic shoot out scene near the end, Travis approaches Matthew and begins conversation. Matthew becomes agitated and kicks Travis leading him getting shot in the abdomen. As Matthew falls you can see a wire on his left side dangling below his thigh presumably from the gunshot squib.