Visible crew/equipment: During one of the scenes where an astronaut does an EVA (Extra Vehicular Activity or "Space Walk"), you see him floating slowly from the Pod shuttle towards the Discovery. This was filmed by hanging the pod on the ceiling and lowering the actor from it with a rope towards the camera that's pointing upward. If you look carefully, you can see the shadow of the rope.
Visible crew/equipment: When the astronauts are approaching the monolith on the moon, you can see the reflection of the cameraman in the visor of the first astronaut off the ramp, in the close up of him walking around the monolith. This shot was personally filmed by Stanley Kubrick and the reflection is his own. (00:48:40)
Revealing mistake: When the lead ape-man is attacking and beating the rival ape-man group leader with a pig bone, the bone bends with each blow, revealing that it was made of rubber.
Other mistake: When Dave is deactivating HAL, he unlocks memory blocks one by one by turning a key for every one of them. When he's done the first five memory blocks, from number 6 to 2, he changes the key from right to left hand, and jumps to the logic terminal blocks, unlocking number 5 of those. However, number 1 of the memory blocks slides out. (01:48:20)
Continuity mistake: Frank is running around the centrifuge. He passes the ladder with two bunks to the right and the work station to the left. When Dave comes down the ladder, the bunk is to the left. This error is caused by Frank's last two running shots (the ones where the camera stayed just ahead or just behind him as he ran) having been accidentally reversed in editting. Frank's hair is parted on the wrong side and the sleep tube layout and the body positions within are precise mirror images of the layout seen in all prior and following shots. (00:56:50 - 00:57:55)
Deliberate mistake: When Dave is in the pod arguing with Hal to let him back in, various patterns of light are projected on Dave's face presumably from the video screens that say NAV and COM and such. Light on such a screen would diffuse and not project like this. It is likely those screens had rear projection and they merely removed the screens for this effect.
Other mistake: When Frank does his first EVA he presses a button to darken his visor, but the button is marked "TEST ALARM" (visible on Blu-Ray version).
Continuity mistake: When Dave begins to disconnect HAL's circuits, one angle shows him unlock the first, second, then third unit. Just as he's about to remove the key from the third lock, the angle changes and he's pulling the key from the second lock and then goes for the third again. We know this is consecutive time as HAL is pleading with Dave the entire time. (01:51:45)
Revealing mistake: The classic scenes of ships in earth orbit is quite realistic, but the Earth lacks an atmosphere. Examining actual photos shows a soft glow around the edge of the earth, missing in the film.
Continuity mistake: When we see the outside of the cave where the apes are sleeping for the first time it's at sunset. We can see the sun and cloud behind the cave as if it is just sitting in a rock face with nothing behind it, but in the next shot when it's daytime suddenly there is a background of rocks and mountains behind the cave.
Deliberate mistake: On the space station after Dr. Floyd has just gotten off the elevator and is talking to a receptionist, the moon image in the window is revolving with the exact appearance it had when Floyd was in the telephone booth when the camera angle is perpendicular to the window and presumably parallel to the station's rotational axis. In the receptionist scene, the view of the window is at an angle yet the rotating moon image is the same. This would require a different axis of rotation for the space staton - one in which the station would be wobbling.
Revealing mistake: When Dr. Floyd arrives at the Moon and goes to the Monolith site at Tycho Crater, the moon shuttle never casts a shadow on the lunar surface. This is especially noticeable in the first shot, wherein the shuttle is coming straight at the camera, the Sun is on the far right, and there is a giant lunar cliff on the far left. No shadow of any sort.
Factual error: In the title shot, the camera rises above the dark side of the moon, revealing the crescent of the Earth, which in turn reveals the full disc of the Sun. The Sun is surrounded by the pinpoint lights of distant stars, but there are virtually no stars visible on the darkened far left and far right sides of the screen. This is exactly opposite of how real-life astronauts describe the star scape: Astronauts say that no stars are visible when looking in the general direction of the Sun, and that stars only become visible to the human eye as you turn away from the Sun.
Continuity mistake: When Frank Poole and Dave Bowman are discussing with HAL the possibility of HAL being in error predicting the AE-35 Unit fault, the 10x3 grid of buttons on the console in front of them is lit differently depending on whether the shot is from behind the men, or in front of them. (01:20:40 - 01:23:00)
Continuity mistake: In the scene where HAL is reading the lips of Bowman and Poole through a pod window, while they are in the pod discussing his possible disconnection, they are shown to be directly facing each other, keeping their heads perfectly still, while HAL looks back and forth from mouth to mouth. In the previous scene, the two men were not conversing in this unusual manner.
Visible crew/equipment: In both EVA scenes, first when Dave retrieves the AE-35 Unit, and when Frank goes to re-install it, there are shots of Discovery in the background, the pod in the foreground and the astronaut floating between. And in both of these shots, the pod rotates 180 degrees. In the front window of the pod, we can see reflected images of the film set and equipment as the pod rotates.