When McCoy, Spock, and Scotty seem to be mesmerized by the women after they are beamed aboard, a close-up of McCoy shows him wearing his medical smock. All other shots of this scene show him without the smock.
When the women are first brought aboard the Enterprise, Ruth's position and body attitude change without explanation. She begins significantly behind the other two women and looking over her shoulder. In the next shot, she is shoulder-to-shoulder with them and facing in the same direction. In the next shot, she is back to her original position and attitude.
When Spock is escorting Mudd and the ladies from the transporter to Kirk's cabin, a blue-shirted crewman suddenly appears in the corridor between shots.
Part way through the episode, the ship depletes its lithium supply and is running on battery back-up. During the external shots following the Warp Nacelles are dark, since they are not capable of operating. When the ship arrives at Rigel 7, the Warp Nacelles are once again glowing, yet they have not received any lithium.
Harry Mudd is wearing an earring on his left ear. However, when they bring up his official criminal record, the photo of him shows the earring on his right ear.
The bridge crew speaks of orbit as if it's something that requires effort to maintain, and Spock notes that they can sustain orbit for 3 days and 7 hours, given their remaining battery power. If they had power enough to enter orbit in the first place, they could enter a stable orbit, which would sustain itself for some time since the gravitational and centrifugal forces would cancel each other out. However, with something as large as a starship, orbits would decay over a period of time.
At the end, when McCoy is on the bridge, he puts his left arm up to his chest and you see a pinky ring on his left hand.
Mudd's Venus drug is amazing since it smooths over wrinkles, fixes your hair, and can even give your face a full makeover, including false eyelashes.
In Mudd's mugshot, he was wearing the same outfit that he is currently wearing in the interrogation room.
It is proven at the end that wishful thinking is what makes a woman beautiful. This happens after one of the women takes a replica Venus pill, which Kirk then reveals to be colored gelatin.
When Harry Mudd contacts the miners on Rigel 12 by means of the communicator, Lt. Uhura should have noticed the unauthorized communication. Additionally, the planet the miners were on was well out of the communicator's range.
In one of the scenes in which Mudd and his women are being detained in a room, he tells each of them how their dreams can be fulfilled; he then boasts that he'll be running the Enterprise. "Captain Kirk, the next orders you'll be taking will be from Harcourt Fenton Mudd." The problem is that there are two security guards right in the room with them. He boasts of this loudly, yet security doesn't report that he plans to take over.
If the Enterprise was operating on battery power only, it's doubtful there would have been enough energy to operate the transporter at Rigel 12. (Of course, when this episode was filmed, the shuttle craft had not been "invented" yet.)
Mudd's height is stated in feet and inches. Earth, and Starfleet, has long switched to the metric system in this century, although they still seem to use Fahrenheit for temperature.
When the ladies are first brought aboard the Enterprise, they are remarkably calm and composed, considering they were just beamed off a ship a split-second before it was destroyed.