Continuity: In the scene where Floyd and his team are heading to the Lunar excavation site, they start to eat those synthetic sandwiches. One of the shots shows a closeup of Floyd. As he is right about to take a bite, the camera cuts back to a shot of the three men. However, Floyd's sandwich is instantaneously about a foot from his mouth and he is fully chewing.

Continuity: Dave only takes one utensil on his tray, later he has two.

Visible crew/equipment: In the scene where the astronauts are approaching the monolith on the moon you can see the reflection of a cameraman in the visor of the first astronaut off the ramp in the close up of him walking around the monolith.

Audio problem: On Dave's deathbed, sometimes the breathing matches his chest movements and sometimes it really doesn't.

Factual error: In the sequence of the shuttle approaching the moon the pilot talks to Dr. Floyd. Floyds food-tray moves away from his knees because of the zero gravity in the shuttle. But it doesn't drift straight away, it swings like a pendulum, so it`s obvious, that the tray hangs on a string.

Factual error: While Dave Bowman & Frank Poole are in the pod bay they lean against the workbench much as anyone would on Earth, yet the pod bay is a weightless environment.

Other: 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.

Revealing: When all the apes are gathering around the big block at the beginning of the film, look carefully at the sky. The clouds never move, not even slowly.

Visible crew/equipment: During one of the scenes where one of the two astronauts do 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 cieling and lowering the actor from it with a rope towards a camera that's poining up.  If you look carefully, you can see the shadow of the rope.