Despite the fact that throughout the dining room scene characters come and go through an interior door, as is clearly shown in later exterior shots, the dining room is set apart from the body of the Cygnus atop a scaffold-like tower with no elevator, walkway, ladder or any means to enter or exit.
After Reinhardt tells Booth about what would have happened if Columbus had been recalled just before he discovered the new world, and that he (Booth) would not even exist, Reinhardt stands and points at Booth with his right hand. In the next shot, a different camera angle, he is pointing at Booth with his left hand.
At the very start of the movie, when Vincent announces, "The largest black hole I have ever seen, Mr. Pizer," and Pizer replies, "Hmm. Let's look at it on the holograph," the viewer can see someone's hand turning Vincent by his left foot. The hand stays there for a moment before pulling back off-camera.
Once the meteorite crashes into the Cygnus, the Palomino crew should have been killed within seconds due to loss of oxygen. However, they survive for several minutes and are even seen scrambling out into the vacuum of space to board the probe ship without any protective suits.
When the "Palomino" is climbing up toward "Cygnus" from the near vicinity of the black hole, it is not in free fall. Therefore, Mr. Pizer would not float up out of his seat when he attempts to go after V.I.N.CENT.
When the Palomino crew are trying to escape to the probe ship, they clearly go outside the structure (and atmosphere) of the Cygnus, as evidenced when Charlie starts to float away. As such, they should die within a minute from asphyxia, but they are able to move around outside for several minutes then clamber into the probe ship, alive.
When the Palamino crew gets on the track car that takes them to the bridge of the ship, Vincent floats ahead of the car, but just as the car starts to move, you can see a metal bar connecting the car to Vincent's back that isn't visible in other shots.
During the robot funeral, only the floor and far wall were built as a physical set. The sloping side walls were a matte painting. At the beginning of the scene, the robot's head at the far lower left disappears under the matte painting. At the end of the scene, when they are filing out of the room, the robot's head at the lower right bobs behind and under the matte several times.
During the dinner sequence after Reinhardt excuses himself to meet up with the probe ship, the film runs in reverse on an exterior shot of the dining room. You can clearly see Captain Holland walk backwards around the room and sit down.
On the bridge of the Cygnus just after Dr. Reinhardt orders Maximilian to stand down, the movie cuts to the seated Dr. Reinhardt. Just after that when Dr. Reinhardt says "Miss McCrae, you may come closer" you can see part of the star field inside his hair just above the hairline.
When the sentry robot pounces on the Captain during the fight to save Kate, the robot does not have his metallic exoskeleton on over his legs (he's merely wearing red leggings). In the next shot, his regular robot shell is visible over his legs again.
The goof item below may give away important plot points.
When the crew flees Reinhardt and his drones during the meteor shower, their open-air 4-seat shuttle is forced to stop as it reaches a ruined segment of track. But the car goes a little farther than it should have and the front of it disappears under the matte painting of the ruined segment.