In the scene when Sam is working on his model and the other "Sam" is banging a ping pong against the wall, he lays his racket on the table. A few shots later a small wooden house appears beside the racket, that no one put there.
After Sam crashed with his Moon Rover, he has a wound near his right eye. In the following scene, when he puts his helmet back on, the wound is near his left eye. Much later, when he's back at the base, it's back on his right side again.
When Sam first approaches his clone while the clone is jumping rope, he has his baseball cap's bill toward the front. Then the camera cuts to a rear view and the hat is backwards. When the camera returns to the front view, his hat is forward again.
When the rescue crew arrives, their spaceship lands partially on top of the lunar station. When second Sam launches his helium pod just a few seconds later, however, the whole station is visible and there is no spaceship on top of it.
Towards the end of the film when the H3 canister system launches Sam into space the shot of the base is the same one from earlier in the film as the Eliza crew ship is not visible in this shot yet when it landed it was clearly almost as big as the base.
After the fight scene, when second Sam drops first Sam out of the headlock, the first Sam places his right hand on the ground, and it does not have the bandage that he was wearing moments before. Seconds later it's back again. Before the drop first Sam's palm full of blood can clearly be seen, before he brings his hand to his nose, and his nose is not bloody until he does.
As revealed in the DVD commentary with director Duncan Jones and producer Stuart Fenegan, the close-ups of GERTY are in fact a prop and the wide shots are post-production CGI inserts. It's clearly visible that the CGI GERTY doesn't have the splashed coffee stains down his right side above the mug holder.
When the two Sam clones are going to put the first clone back into the crashed Rover, the Rovers are both facing front-end outwards in the bay. However, when the injured clone drove back after locating the aerials, he drove in front first, while the other clone reversed in. So the Rovers should be facing different directions.
The mug on Gerty's cup holder appears and disappears throughout the film. Sam leaves his mug on top of the "Mark" terminal at the start of Gerty's first scene, for example, yet it is on the cup holder as he walks down the corridor.
When first Sam has fallen asleep and begins to dream about his wife, the "Take Off" magazine on the shelf beside his bed is facing away from him. When he wakes up, the magazine has been turned around 180 degrees.
(at around 25 mins) When the two Sams are looking for the towers, the interior of Rover #3 shows hanging fuzzy dice. These dice are actually from the crashed Rover #1, and are still there in the ending shots.
Sam realizes that he has lost a tooth when feeling around in his lower jaw. However, the tooth that he pulls out of the toilet has three roots, meaning that it is a maxillary (upper) molar. Mandibular (lower) molars only have two roots, unless the person is of Asian or Native American descent.
They are harvesting H3 on the far side of the moon, but a few times the Earth can be seen in the sky near the base. Since the moon does not rotate (relative to earth) there is one side where the Earth is always above the horizon (the side facing earth) and one side where the Earth is always below the horizon (the side that points away). The Earth cannot be seen from any part of the far side. The Earth is always below your feet.
When Sam makes a phone call to his daughter, there is no noticeable lag time between responses. Radio transmissions take about three seconds to make the half million mile round trip from Earth to the Moon or vice versa.
As Sam enters the code to access video files, a quick shot of the keyboard shows that he is only addressing keys on the home-row (and alternatively hitting the 'j' and 'k' keys). Although a password may consist of just these keys, it is highly improbable. Also, the characters appear on the monitor before he has a chance to type them.
The last time we see Sam Bell in the pod (shouting "whoohooo!") the pod is clearly rumbling and shaking. This can either be vibration of the engines or by reentering the atmosphere. But the shot directly before shows the engines turning off and earth's atmosphere is still far away. In that situation there's nothing that could rumble or shake the pod.
GERTY is a computer that has robotic appendages to allow it to interact with items in the base and to provide an interface for Sam to communicate with. As a computer, GERTY would always be in direct communications with the command center on Earth, and all of the rest of the equipment in the base (as well as any other bases on the moon). Thus GERTY would not communicate with Earth through video and voice link as if he were a person. He would transmit digital information directly and receive commands directly, like any other computer-controlled equipment. However, since the writers needed some way to indicate to Sam that the long-distance communication was actually functional, GERTY needed to use video communication so Sam could overhear and see the long-distance link in use.
Lunar regolith that has been turned over by harvesters is shown as darker than untouched regolith, but in fact newly exposed regolith, for example that which has been exposed by meteorites, is bright, becoming darker with age.
The name of the Korean company that owns the Lunar Station (Sa Rlang) is mispronounced several times throughout the movie. The correct pronunciation of Sa Rlang is with the "a" in both words having the sound of the "o" in the English word "odd".
Helium is an inert gas and does not participate in chemical reactions. However, it can participate in nuclear fusion. The isotope helium-3 is particularly interesting to fusion researchers, and is more abundant on the Moon than Earth.
After Sam makes a call to his daughter we can see his vehicle and Earth above horizon in one shot. This may seem a goof, since base is supposed to be on the far side of the Moon, which is invisible from Earth and vice versa. However, Sam needs to get far from the jammers to be able to call Earth; explaining why he is on the other side of the moon.
Sam and Overmeyers both say "over and out" at the end of transmissions. While this is procedurally incorrect ("over" means "waiting for your response" and "out" means "ending transmission"), most people don't know any better.
When the two Sams find the "secret room", they enter and leave using a hidden ladder. They use that same ladder to take out another clone, to use on their plot to escape. How does GERTY get the clones out of the nursery?
When second Sam gets into the helium canister launching pod he doesn't have to press any buttons to close the door on the pod, or type in the code to launch it. GERTY can't do this because Sam rebooted GERTY.
There are many shots outside the station where sunlight and stars are visible at the same time. Even without an atmosphere this is not possible because the contrast between the bright ground and the dark sky is too high.
When Sam is walking outside it looks like the moon's gravity. But inside he moves normally, although his weight should be only one sixth, assuming there is no sort of 'artificial gravity' being used/supplied in the indoor areas.
Furthermore, a game of table tennis would look very different under the gravity of the moon.
Probably due to the film's low budget of $5 million, there is no attempt to correctly depict the gravity on the moon (one-sixth of Earth's.) As an obvious example, when Sam is playing ping pong, the ball reacts as it would on Earth.
The goof items below may give away important plot points.
When the two Sam clones are going to put the first clone back into the crashed Rover, the second clone puts the first clones helmet on and we can see that the visor is dusty and bloody. In the next scenes the visor is spotlessly clean.
When the younger Sam is looking for the secret door, he gets in a fight with the older Sam and eventually flips over his "model" of his hometown. The model is destroyed. However, a few scenes later when the younger Sam is apologizing to the older Sam, the model is not destroyed and clearly visible in the background.
When Sam is marking his "time" with the faces in dry-erase marker, you can see the outlines where previous dry-erase marker has been wiped away, leaving a negative image of the smiley face in its stead. However the base is "reset" and previous markings wiped away at the end of each "shift", so these residual markings are likely if not expected.
In the final camera shot from the base, a "Pause" indicator is briefly visible in the top-right corner of the main screen, just before the communications come online and the "Live feed" becomes available.