Many of the continuity errors are not in fact errors. Stanley Kubrick purposely included many continuity errors as a way of creating a feeling of disorientation for the audience. That is why people's positions change, props are reorganized, and hats (and other articles of clothing) appear and disappear.
There is a running theme of the Government referring to Citizens and the State, implying that Great Britain has abolished the monarchy in this "future". Yet the prison is called "HM Prison Parkmoor", HM being a common designation for Her/His Majesty.
Alex comments that the opera singer's voice makes "all the malankey little hairs" on his skin stand up. In the Nadsat slang that he speaks, "malankey" already means "little", so his statement becomes one about all his "little little hairs".
Just before Alex and Miss Weathers fight in the health farm, the room is full of cats. During the fight, the panning camera reveals that there are no cats in the room. The cats mysteriously reappear immediately after the battle.
When Alex was reading from the Bible, and asks the priest to talk with him in private, we see the priest putting his hand on Alex's shoulder, while in an another cut we see him preparing him self to put his hand on him.
The sandal of Billy Boy's hostage, that laces up her leg to the knee, changes position for right leg, to left leg and back to right again as she runs off after Alex and his gang disturb the rape attempt.
(at around 33 mins) The water of the flat lot marina changes in turbulence several times, low waves at first and then less choppy while Dim is falling and finally completely still as he hits the water.
In the marina scene, George is seen wearing his hat as he gets up in the shot of Dim falling into the water. Moments later as Alex is cutting Dim's hand, George is doubled over in the background without his hat. As Dim falls backward in recoil from the cut, George is seen in the same position with his hat on.
At the abandoned casino Billy-Boys' gang attack a young woman.
Her sandal, that laces up her leg to the knee changes position from right leg, to left leg and back to right again as she runs off after Alex and his gang disturb them.
The writer's typewriter appears to have perhaps ten or more sheets of paper loaded on the roller. Even if you were making carbon copies, the limit would be three sheets of paper, and two thin sheets of carbon paper. There are no signs of any carbon paper being used at this desk however.
At 19m 45s Alex's mom is supposed to be shouting at him outside his bedroom door. She appears to shouting at the inside of a front door (two locks and a handle), but it is later revealed that Alex has installed extra locks on his bedroom door (including a combination dial lock).
The wine Alex drinks appears far too light in color to be a 10 year old Medoc. Stanley Kubrick did use the actual wine described by Alex (specifically a 1960 Château Saint-Estèphe Médoc). But after each take water was added, ostensibly to keep the wine level consistent without intoxicating Malcolm McDowell. By the time Kubrick got what he wanted the wine was visibly watered down.
During the high speed car ride, the trees and greenery on both sides of the road are illuminated. As the viewpoint is looking backwards from the front of the car, everything that we see is behind the vehicle, and as there are no streetlights it should all be in darkness.
The goof items below may give away important plot points.
When Alex jumps out of the window, the camera view gives the impression that we are seeing Alex's view of the onrushing ground. However, he does a flip, winding up with his back pointing to the ground so the camera angle is inconsistent.