When android Norman first comes on board the Enterprise and starts to take over the Engineering section, there's a red-shirt crew member who's presumably about to dive from the upper platform area onto Norman, but he's going in a direction AWAY from the android. Yet in the split second jump cut, he lands directly on top of Norman.
When the Enterprise enters orbit around Mudd's Planet there are no rings visible around the planet. However when the Enterprise leaves orbit at the end of the episode, large rings are in evidence.
When Kirk and the party first meet Harry Mudd, there are two Alice androids flanking Mudd (#99 and #66). In subsequent shots, the Alice on Harry's right (#66) disappears and reappears several times.
When Norman enters Engineering and Scotty is slugged, Scotty falls back and clutches the end of the control console near the comm station. A close-up shows Scotty warning the Bridge from that station. But in the next wide shot of Norman dispatching the rest of the Engineering staff, Scotty is not there. We then get another close up of Scotty passing out at the comm station.
In the teaser, Norman makes it from Engineering to the Bridge in under twenty seconds. While the exact location of Engineering was never established in the original series, no secondary sources have ever placed it that close to the Bridge, and logically it should be closer to the warp engines than the Bridge.
In the closing credits, the twins Alyce Andrece and Rhae Andrece are credited as "Alice #1 through 250" and "Alice #251 through 500," respectively. But this is incorrect as generally when two Alices are visible both their numbers are below 250: #66 and #99 with Mudd on his throne, #27 and #210 with Spock in the control room, etc.
Chekov says he's been in Leningrad. The show's writers did not anticipate that the Russian city of St. Petersburg would drop the name "Leningrad" centuries before Mr. Chekov was born. But being unable to predict the future is not a "goof."
When two of the female androids start to speak in unison while talking to the crew, one of them says, "You are free to use them," while the other one says, "You are free to visit them."
In the beginning, when Norman is furiously flipping switches in Auxiliary Control, he, at one point, attempts to turn a clearly stationary light.
Obvious stunt doubles when the android takes over Engineering and fights with the crew.
Acne scars on Norman, an android.
If it's that easy to hijack the Enterprise simply by taking over two isolated rooms, those areas should have better security, such as a futuristic Cypher lock or retina scanner, than just a warning sign stating "authorized personnel only".
When Kirk attacked Harry, one of the Alice's stood by and did not defend him despite their program to serve "Lord Mudd".
When the crew first beams down to Mudd's planet, a stagehand's feet can be seen backing behind the door frame to the rear of the landing party.
The color shirt of the android who takes over the ship is blue. He keeps skipping medical appointments and Bones is concerned. However, the blue shirt is Science and Medical, which means that if anyone is transferred to the ship, both Spock and Bones would have received paperwork on the individual. They both should have known who the person was and what their purpose was on the ship. Yet, in the opening scene, Spock doesn't know him at all and Bones can't get him to keep a medical appointment.
It is never explained how Norman got posted aboard the ship to begin with. The ship's personnel officer (seen in Court Martial (1967)) would presumably have to verify his orders and background with Starfleet Command.
When Norman is on the Enterprise, he is not wearing a communication necklace, which implies the necklaces are not necessary for androids to function. However, it seems to be required when an android is processing incoming information or responding to requests for information - particularly illogical or disjointed information.
With all of the crew transported to the planet surface, how did they get transported back to the Enterprise at the end when there was no-one on board to operate the controls?
With a planet of over 200K Androids at his disposal, Mudd would not have needed this elaborate scheme to have a Starship brought to the planet. The Androids could have created one for him, or at least a working ship that Mudd could have used to eventually leave the planet. It is the androids that stole the Enterprise, they used Mudd to facilitate their plot. The androids plot was to steal the Enterprise and spread themselves throughout the galaxy to "serve" and control humans.
The androids are not supposed to have any emotions, yet the two Alice androids appear to be visually offended when Kirk tells them he doesn't like them and shoos them away.