A detail shown but not commented on is that the Kaldorean spaceship does not maintain an internal atmosphere while the crew is in suspended animation. Simmonds should have suffocated in his sleep as the oxygen ran out. (Note: Simmonds had been in suspended animation, as the others had, and wasn't breathing oxygen until he revived, then did indeed suffocate after thrashing about.)
In the opening scenes in Main Mission, which take place in real time, it takes Alan Carter less than 40 seconds to put on a spacesuit, get to an Eagle, and have it ready on the launchpad. A total impossibility considering the preparation required and the size of the moonbase.
Koenig is summoned to the Kaldorean ship to find Helena unable to be brought back to consciousness during the suspended animation test. The situation seems critical as we go into a commercial break. It is not explained why the Commander is wandering around Main Mission at the start of the next scene instead of remaining with Helena during the crisis.
When Simmons fails to contact Earth, he checks his commlock, sees the time, and immediately concludes that the suspended animation process didn't work on him. Looking at a clock isn't the same as looking at a calendar. His reaction makes no sense because he didn't check the actual date. (Note: Simmonds discovered his comm-lock was registering "Lunar Time", indicating he was still within communication range of Moonbase Alpha; and according to the time, the ship left Alpha just a few hours ago. Since the Kaldoreans were still immobilized, he correctly concluded that the process didn't work for him.)
Two of the hibernating aliens can be seen moving in response to Simmonds beating on his chamber. This was due to Roy Dotrice (Simmonds) unable to hear the director yelling "cut" while banging on the cabinet. Despite the background actors breaking character, Dotrice's animated performance was included in the final cut.
Before Simmons is about to take up his travel position in Zantor's craft for the voyage back to Earth, he asks Zantor (Christopher Lee) to go first. As Zantor lies down you can clearly see his long white wig fall off revealing Lee's natural dark hair.
The special effects technicians had a difficult time maneuvering the alien ship. When it's shown landing on the surface of the moon, the thrusters made the model unsteady, then when it's about to hit the ground, you can see the guiding line just for a moment. It is vertical and attached to the landing gear nearest the viewer.
When Simmons is frantically beating on his transparent chamber, there is an over head shot where he collapse from exhaustion. There is a gap of space along the right border of the chamber. For the purposes of the story, he's trapped, but technically he could have gotten out.
While the Alpha team is on the alien ship, the Moonbase team pulls up visuals of them on a screen. But since no one on the ship is shown with a camera, it isn't explained how that happens.
The graph displays never change on the green display screens of the X5 computer panel.
When Simmons takes Koenig's comlink, Koenig tells Kano to have the Computer program a new one and, "Explain to the Computer as best you can." Why does Kano essentially need to explain making a spare?
A string or stings can be seen briefly controlling the first landing of the alien spacecraft.
When Eagle One takes Koenig and co to investigate the alien spacecraft on the lunar surface, it leaves the Eagle passenger module at the spacecraft while Alan Carter flies the Eagle back to the moonbase. There is no logical reason to do this. Since the Alphans don't yet know who or what is inside the alien ship, they may need to get out of there quickly, but their only means of escape has just flown away. Obviously, the producers were just trying to show off the "wow factor" modular nature of the Eagles.
The Kaldorians have been en route to Earth for over 300 years and have another 75 years to go on their journey there. The moon left the Earth's orbit quite some time ago and is wandering through space so far away the Alphans don't even know where Earth is now, having gone through space warps and a "black sun" from which it emerged on the far side of the universe. It is a preposterous contrivance that the Kaldorian ship could find the moon, either by accident or intentionally.
Commissioner Simmons takes the base's primary power converter as hostage in order to gain passage on the alien ship, and in the ensuing conversations it's identified as a one-of-a-kind, irreplaceable component. Such an essential piece of equipment would have to have spares or replacements, since it would eventually break down or wear out from being used.
When the Alpha team are on the alien ship, Koenig tells Alpha Command to add atmosphere and artificial gravity to the ship. But it's never explained how that's possible from a distance. (Note: Evidently, a Docking Eagle possesses special functions to accommodate a docked spacecraft for the arriving team).