Corrected entry: When Riker's father is talking to Troi, he says Riker has just been offered his first command (the USS Aries). However, Riker was already given a offer to command the USS Drake, which he turned down to join the USS Enterprise. This would be at least his 2nd offer to command, something his father would be aware of given his position, and following his son's career has he has.
The Royale - S2-E12
Corrected entry: When Data is trying to talk to the elderly lady playing the slots, she is dropping coins and pulling the arm in much too rapid a fashion to allow the wheels to come to a stop. Also, she's continuing the motions as you hear the machine paying out.
Correction: The Royale is not a real place with real people or real slot machines. It was created by an alien race with one dimensional characters and the old woman is just a cliché background character that doesn't interact with anyone. Her actions are completely constant with a poorly written book being interpreted by aliens (such as "an old woman wasting time playing the slots, pulling the handle over and over until she wins and never stopping after.").
Corrected entry: As Wesley and Picard prepare to depart the Enterprise, Data says the ship is at impulse speed. However when we see the shuttle exit, the ship is shown not to be moving.
Correction: At impulse speeds, there would be no visible indication of movement as they're in open space.
This is incorrect. In-universe has shown the difference between a moving ship and stationary ship, even open space.
Exactly what is this difference you're referring to? If a ship is moving at sub-light speeds with only stars visible it is impossible to tell how fast it is moving, or even if it is moving. The only movement visible would be that imparted by the difference in movements of the camera and ship. (In fact, they create of the illusion of the ship speeding by by moving the camera past a stationary ship.) This is reality. If there is an error, it is the fact that the stars can be seen moving outside the hanger bay door which would indicate the ship was slowly rotating.
Corrected entry: Data takes a holographic paper off the holodeck.
Correction: It is often explained that the holodeck uses a combination of holographic projection, transporter, and replication technology to achieve its illusions. This is why people on the holodeck are able to eat food created there. Something like paper would be easily replicated as real and so could leave the holodeck. Living beings are much more complex.
The Dauphin - S2-E10
Corrected entry: After viewing the clash between the governess and her protector, the camera shows Wesley. His communicator badge is completely silver, instead of the standard gold and silver. (00:34:55)
Correction: This is not a mistake. Wesley's entire combadge was silver (arrowhead AND oval) from the beginning of Season 2, when he received the provisional officer's uniform. He did not receive a gold-oval combadge until his full promotion to Ensign, which also brought a standard uniform.
Where Silence Has Lease - S2-E2
Corrected entry: Deanna is originally sitting on the bridge when they approach the zone of darkness, but after Picard tells Wesley to take them in, in the next wide shot Deanna disappears entirely and doesn't reappear throughout the rest of the scene - she only reappears much later when Riker and Worf return.
Correction: Wesley also says it will take 12 minutes to intercept the "hole." When Picard says let's launch a probe, Deanna is already gone. Picard then says this is close enough, indicating they've intercepted it. So almost 12 minutes have passed, plenty of time for Deanna to leave the bridge for a number of reasons that are simply not addressed on screen.
Corrected entry: When the head Bringloidi starts a fire in the hold, Lt. Worf informs him that a force field will cover the fire until it is extinguished. But in many later episodes you see fires burning, and nothing happens. At most, they will let the air out of the room, but no force fields.
Correction: The fire suppression systems are known to fail when a ship has taken a lot of damage. There has even been an occurrence of an officer stating, "Fire suppression systems are not functioning."
Corrected entry: The Stargazer model in Picard's office changes position each time someone visits him - facing right, then left, then right again.
Where Silence Has Lease - S2-E2
Corrected entry: In this episode, the number of the USS Yamato is given as "NCC 1305 E." However, in the episode "Contagion," from later in season 2, the number is given as "NCC 71807."
Correction: The Yamato seen in When Silence Has Lease is not the real ship, but a replica created as part of a test for the Enterprise crew by the Nagilum - as part of the testing process, it contains numerous inaccuracies, such as being made from the wrong materials and has many alterations to the layout of the regular ship, including multiple versions of the bridge. The incorrect registry code is just one of these differences from the original. Later in the season, in Contagion, we see the real ship, which obviously carries the correct registry code.
Corrected entry: When Data hands Geordi the paper with the image of the Enterprise on it, Geordi looks at the picture then turns it over. However, when he flips the picture it is the right way up, so it means that Geordi has been looking at the picture upside down.
Correction: Geordi is intimatly familier with the Enterprise. He can recognize it no matter how it is oriented.
Corrected entry: When Pulaski says "Your omelettes deserves nothing less", Riker is actually serving Pulaski, Data, LaForge and Worf scrambled eggs. (00:00:40)
Correction: She does say that, however she says it prior to Riker cooking the eggs. For whatever reason she may have been expecting omelettes and he simply made scrambled eggs. If she had said after seeing the scrambled eggs, it could be considered a mistake.
Corrected entry: In the shuttle Picard picks up the plate of sandwiches with his left hand while holding a cup with the right. After Wesley takes a sandwich, the camera cuts back to Picard, who has the plate in his right hand, and no cup to be seen.
Correction: Picard never holds a plate of sandwiches and a cup. He has the plate of sandwiches in his left hand and a 2nd, empty plate in his right hand. He gives the empty plate to Wesley as Wesley takes a sandwich. Picard then transfers the plate of sandwiches to his other hand so he can take a sandwich with his left hand.
Correction: He says that he has been offered his first command, not that it was his first offer of a command. If Riker turned down ten offers, the 11th would still be an offer of his first command since he doesn't have one yet.