(at around 1h 30 mins) When Simon is shooting at John Spartan during the car chase scene, he shoots out of the back of the car. There is no sound of the rear window glass breaking, and it never shows him breaking the rear window previously.
(at around 44 mins) When Phoenix punches (and kicks) the glass to try to take the guns, the display glass sounded like it was plastic, but when he threw the guard into the display glass, it shattered, like it really was glass.
(at around 41 mins) When you hear the voice of Dr. Cocteau narrating the museum exhibit, he states that below the museum patrons is an actual 20th Century street preserved since the earthquake of 2010. If the earthquake happened in 2010, the preserved street would be from the 21st Century, not the 20th.
(at around 45 mins) When Simon Phoenix is in the museum querying the computer about the "linear accelerator gun", the computer voice states that the gun achieves fusion in 2.6 minutes. However, the computer screen indicates "fission time 2.6 minutes" on the diagram of the weapon.
(at around 1h 26 mins) When introduced, Edward Friendly is all about "... I like to think, I like to read. I'm into freedom of speech and freedom of choice. ..." Yet, once the Cocteau regime is overthrown (at around 1h 45 mins), he tells Associate Bob to change his attire and hairstyle.
Lenina Huxley has a great interest in 20th century history and culture and has a collection of vintage posters like Lethal Weapon 3 in her office and other things considered to be contraband, yet her favourite music is commercial jingles and TV themes instead of actual music.
Several times during the movie, Simon Phoenix's shoulder pads (one side resembles an ATV tire) switch sides. Not only that, but Phoenix has two different color eyes - left blue and right brown - which also switch sides.
(at around 3 mins) In the beginning when Simon finishes using the dagger to puncture the gasoline barrels, he lays it back on the table and it is dry. You can see the gasoline spraying all over as he jabs each plastic barrel. He would also have the gasoline on his hands when lighting the cigarette.
(at around 11 mins) The display on John Spartan's cryogenic reads, "August 3, 2032", which will fall on a Tuesday. Later that same day, Huxley says to Warden Smithers, "It's a beautiful Monday morning".
(at around 10 mins) Spartan's body position and mouth change several times while he is supposedly frozen. Before he freezes, his mouth is closed and his body turned. When he freezes, his mouth is open and his hands placed facing up (showing surprise, etc.). The levels of surprise shown in the mouth and body position change throughout.
(at around 42 mins) In one scene Spartan changes seats with Huxley in a police car. When Spartan sits on the driver's seat Huxley can be clearly seen in the background. But after they switched their seats Spartan doesn't appear behind her.
(at around 26 mins) When Phoenix first gets out he makes his way to a computer and the police come to arrest him. Phoenix takes one of the policeman and pushes his head through the side glass pane and walks away to fight the other officers, but when Phoenix goes back over to the computer the glass is not broken.
(at around 1h 35 mins) When Spartan and Huxley arrive at the cryogenics place to battle Phoenix and those surviving CryoCons, Phoenix sees them on the monitor walking into the building and Spartan is on the left and Huxley on the right. In the next shot, Spartan is on the right.
(at around 27 mins) When Phoenix is fighting the police at the information/teller machine right after he escapes, the fourth cop he engages gets his sunglasses punched off. They change the angle and he has them on again (as he gets shoved to the ground).
(at around 1h 35 mins) When Phoenix's goons shoot Cocteau and throw his body in the fire, it is obvious that it is a life size doll that was thrown in the fire as it was way to rigid to be a human body.
(at around 58 mins) Huxley states that because salt is harmful, it has been made illegal. In truth, the minimum requirement of sodium necessary to sustain life has been estimated at 500 mg per day; in other words, without salt, you would die. If you're active and therefore perspire a lot (as is perhaps the case with John Spartan), your need for salt increases significantly.
(at around 5 mins) When Spartan goes against Phoenix in the beginning of the movie, there are barrels lined up along the wall labeled 'C-4' (Composition 4, aka plastic explosive). C-4 is not stored in barrels, and C-4 will not detonate because of fire (as is suggested). If you set C-4 on fire, it burns just like wood - it needs to have a detonator implanted and triggered in order to explode.
(at around 1h 14 mins) After "having sex" with Lenina Huxley, John Spartan returns to his apartment, picks up a ball of yarn, and starts winding the yarn around his hand and elbow to create a skein, which makes no sense. Yarn which is sold in skeins is often wound into balls prior to use to avoid tangles, so Spartan's ball of yarn is ready to go; winding it into skein serves no purpose, other than increasing the likelihood of potential tangles, which any serious knitter would want to avoid.
(at around 19 mins) When the computer is responding to the first Code 187s, it is heard to say, "Last recorded offense: September 25, 2010," which in the movie would have been 22 years previously. However, when Spartan is first thawed, Garcia tells him that "There have been no deaths of unnatural causes in San Angeles in the last 16 years." His math is off.
It is stated that the last recorded offense of "187" (which is the actual current California Penal Code section for murder), was 22 years previous. It was also stated, "There have been no deaths of unnatural causes in San Angeles in the last 16 years." Both can be simultaneously correct. Death by unnatural causes does not assume murder, and they are not the same. It is possible, and consistent with the story line, although excessively Utopian, there were no murders for 22 years, and there were no unnatural deaths for 16 years. There are countless possibilities of death, including homicide, that are not murder. For example, if a person choked on a cherry pit and died, it would be unnatural, but not (likely) murder. As well, industrial accidents resulting in death occur frequently, and are not murder. It's hard to believe that any society could continue for 16 years with no unnatural death, but improbability does not equal impossibility.
(at around 5 mins) In the beginning of the movie, a stockpile of C4 catches fire, and thus blows the whole building up. C4 only can be detonated by a detonation charge or blasting cap which is just a smaller explosive. If it catches fire, it burns slowly. This is true of C4, however there were no details to describe, and no one suggested, the C4 was only detonated by the fire. It is reasonable to believe Phoenix detonated them by other means. Spartan stated to his captain, "he rigged the place to blow". The complete destruction of the building in successive detonations is a firm indicator the charges were strategically placed throughout, and anyone with that skill would not rely on a gasoline fire for detonation. It is also possible the gasoline fire was intended as a lure for responders, as it would increase the casualty count at the time of the explosive detonation. Phoenix did not expect Spartan to appear at that time, which may have caused Phoenix to alter his plan in any number of ways.
Both Spartan and Phoenix receive verbal morality tickets throughout the film the latter's tickets are issued without giving a name due to his lack of a code chip. By that logic he shouldn't receive a fine at all because in addition to identification the chip is also responsible for all monetary transactions meaning any fine incurred couldn't be enforced.
This is a futuristic computer-operated system in a world where everyone is Lo-Jacked, and no one has reason to believe otherwise. Phoenix affected his own premature release, and was not subject to the "normal" method, which certainly would have included the implant. It isn't unreasonable to believe the computer system automatically triggers on Verbal Morality Code words or phrases, and responds even when it is unclear who is the offender. It would be reasonable for the computer to simply respond as programmed with, "[null], you are fined one credit for violation of the verbal morality code", and print the ticket with no name.
(at around 45 mins) Firearms on display in a real museum would be deactivated so they wouldn't work, and no ammunition would be available. The guns in the movie are all working and surrounded by live ammunition. That's deliberate. In a future society so safe that even the police don't carry firearms, they wouldn't consider deactivation necessary.
(at around 49 mins) When Phoenix falls through the glass at the museum into the LA exhibit, you can see him sliding down a visible safety rope. His feet are even in the position to be sliding down the rope.
(at around 57 mins) Even though the audio dubbing and logos for Taco Bell were changed to Pizza Hut in post-production in some versions of the movie, the editors missed some. The original logo for Taco Bell can still be seen on the windows of the restaurant when Lenina and John go out to dinner. The sign outside clearly says Pizza Hut, while the windows show the Taco Bell logo.
The goof items below may give away important plot points.
(at around 11 mins) When the film first progresses to the future, the date is 3 August 2032. However when the police computer reports Warden William Smithers' murder-death-kill (at around 20 mins), it states the date of death as 4 August 2032.
(at around 1h 40 mins) During the final shoot out in the cryogenics lab, Spartan is firing a single handgun at Pheonix from behind a computer. He then dives to another terminal, now firing two handguns at once. When we cut back to the new terminal, he's now holding and reloading a single gun.
(at around 1h 40 mins) When Dr. Cocteau is killed and thrown into the fire, there isn't enough time for his body to have been reduced to ashes by the time Spartan and Huxley see the remains of his clothes. A common fire place such as that, would only have caused major charring of the skin, leaving a body for them to find.
John Spartan's only conviction was manslaughter, albeit of approximately 20-40 people, while Simon Pheonix's list of convictions was extensive, and yet he was able to get a parole hearing 14 years before John Spartan's was scheduled. However, since Dr. Cocteau owned the cryo-prison and was able to add all of those illegal elements to Simon Phoenix's rehabilitation (weapons training, computer skills, combat training, etc.), he could have also easily arranged for Phoenix to have an earlier parole hearing than would have originally been planned.
John Spartan is incarcerated because his actions supposedly lead to the deaths of Simon Phoenix's hostages; he is confused because his thermal scan of the site did not show the hostages, yet their remains are found after the explosion. At the end of the movie Simon Phoenix reveals that the hostages were already dead before John Spartan's raid - any competent medical examiner would easily be able to deduce from a number of factors (for example, the lack of smoke damage to the lungs) that the hostages were dead before the explosion, thus clearing John Spartan of any wrong doing.