When the Ferrari crashes through the parking meters, the indicator lights on the bumper have disappeared, but in the next shot of the Ferrari from the front, the lights have reappeared.
The windshield of the Ferrari is shattered by crashing into the parking meters. A few seconds later while Goodspeed if fighting to release the airbag the windshield is intact.
When the Hummer crashes into the meter-maid vehicle, it skids and goes sideways. In the next shot, it is still pointed straight down the street.
The same extras (most notably the man with a large white beard) can be seen standing both on the left and on the right of the Ferrari when it breaks the storefront window.
Mason and Stanley dry off too quickly after sheltering underwater from the firebombs.
For operational security reasons, Navy SEALS never acknowledge each other by rank, nor do they use their surnames. Everyone on a SEAL team, even the officers, only address each other by first name.
As the Marines are patrolling, they continuously charge their weapons. While on patrol, it would be assumed that they would already have a round in the chamber. As a result, charging the weapon again would not only be unnecessary, but it would cause the weapon to eject a perfectly good round.
Mason uses the edge of a smashed quarter to score a circle in the interrogation room's one-way glass observation window. The metal of a quarter is too soft to score glass.
VX was not discovered by mistake. It was discovered by Dr. Ranajit Ghosh when he did research on pesticides. It was not a failed pesticide either, it was used as a pesticide until 1955. It was, however, withdrawn due to its high toxicity. From 1961, it was manufactured for weapons use.
During the chase, when Goodspeed crashes the Ferrari into the parking meters, several dozen coins spill out including a good number of pennies. Modern San Francisco parking meters do not accept pennies.
When Mason is with the troops in the basement of Alcatraz, he states that he studied the timing of the boiler that he must roll through. The question has often been asked why he wouldn't then just be able to open the door during his original escape. However, when Alcatraz was functional, the door may well have been guarded or locked from the other side.
Mason has been imprisoned for the past thirty years yet when he escapes he knows how to work a cell phone (reasonable, since it's not hard; he also mentions that he watched TV in jail, so he has probably seen plenty of examples) and modern SCUBA gear (which hasn't changed all that much).
Though it is unusual for the President of the United States to appoint someone as young as thirty-three years old as his Chief of Staff, both Presidents Ford (Dick Cheney) and Carter (Hamilton Jordan) had Chiefs of Staff under 35 years of age.
Right before the Marines betray General Hummel, Captain Darrow refers to Gunnery Sergeant Crisp simply as "Sergeant". NCOs and SNCOs in the Marine Corps are always referred to by their full rank, i.e, "Gunnery Sergeant", never simply "Sergeant". Captains Frye and Darrow are, however, likely Army captains and not Marines (multiple examples are given for this), so they are not as immune to this mistake.
The Ferrari airbag deploys too late and too slowly. Furthermore, it is over-sized.
This is a deliberate choice for comedic effect.
During the chase scene with the Hummer, one of the cars that flips, clearly showing the explosives-holding special effects canister under the car, still smoking.
The old lady seen crossing in front of the crashing cable car is also a passenger on the same trolley.
Rubber tires are visible underneath the cable car as it crashes into the SUV.
When Goodspeed is riding the dirt bike, on a close-up shot of Goodspeed on the phone, you clearly see the front tire of the dirt bike is not moving.
When Goodspeed is talking to Isherwood on the cell phone while riding the motorcycle, on long shots it can be easily seen that Goodspeed is played by a stunt performer.
In the gunfight when General Hummel is killed, a drop of fake blood hits the camera lens.
While Agent Goodspeed is describing the horrible effects of VX gas, the part where he says "it melts your skin off" does not match his mouth movements, and sounds different than the rest of the audio.
When Mason and Goodspeed fall from the cart and then the Marines come down along the track, we can clearly see the rope attached to the first Marine from certain angles.
When Goodspeed is blown away by the bombs, you can see the wires that pull him.
Cameras are visible on the roads at several points during the car chase (Mason's Hummer can be seen to drive over one as it collides with the vehicles by the telegraph pole).
Just before the president gives the order for the air strike the camera shows ground crew marshalling an F-18 coming towards him and turning to its left. As the camera pans down a towing arm can be seen attached to the nose wheel indicating that the aircraft was not moving under control of a pilot but being towed by an unseen tractor off-screen (as at an airport when aircraft are pushed back from the terminal).
The launched missile is said to be launched toward a football game in Oakland. However, the next shot shows the missile flying over Candlestick Park in San Francisco, not the Oakland Coliseum. In fact, the missile is clearly shown grazing trees on top of Bayview Hill.
Goodspeed lives in Washington D.C., but when he calls Carla, you can clearly see out of the apartment window that it is the skyline of downtown Los Angeles.
During the chase sequence, the police report that he is headed "west on California", then the camera pulls back to reveal them heading north towards Coit Tower.