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.
Corrected entry: The "Romulan" that attacks Geordi is not standing in the spot he is in when time pauses again.
Correction: It's actually not a mistake, Geordi realises that the "romulan" wasn't in the right spot, it is as he begins to voice that something seems odd that he is attacked.
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: When talking to Samuel Clemens, Jack London tells him of a dream to visit Alaska. This episode is set in 1889, and Alaska wasn't a state until 1959.
Correction: But it still existed as a place. It had been known and inhabited and explored for hundreds of years. Somewhere doesn't have to be a US state for someone to want to visit it.
All Good Things... (1) - S7-E25
Corrected entry: On the Pasteur, when Data reports another ship de-cloaking, he says "It's the Enterprise." He uses a contraction rather than saying "it is." Even though it's in the supposed future, many times Data has pointed out he can not use contractions.
Correction: Data's use of contractions in the future demonstrates that he has evolved further towards his goal of being more human.
Correction: As you point out, this is in the future. A future where he has mastered emotions. It's assumed he's overcome the contraction limitation as well.
While I can agree that is possible, I find it a bit unlikely. My reasoning is that this occurred in the possible future presented by Q, because he was taking Picard through time. In Generations, he had taken Soong's original design for the emotion chip and improved it. However, he still was not using contractions.
In the episode DataLore, Wesley realises that Lore has taken Data's place because Lore was using contractions and, as Wesley stated, Data never uses them.
Corrected entry: They beam up with the injured Romulan to the transporter room and Beverly says "let's get him to sickbay." They could have just beamed directly to sickbay.
Correction: Within the Star Trek universe, site-to-site transport carries higher risk than using a transporter pad. It is supposed to be used only in extreme emergencies. It is up to the doctor to decide whether or not someone needs to be transported directly to sick-bay, and in this particular situation she deemed it unnecessary. Why sick-bay doesn't have it's own dedicated transporter pad is a discussion for another day.
Encounter at Farpoint (1) - S1-E1
Corrected entry: Wesley walks out of the Halo deck, dripping wet. Things from the holodeck are not real and disappear upon leaving. Wesley should be dry when he passes the threshold.
Correction: Mistakes like these have already been entered and corrected. The holodeck uses transporter technology with replicators to generating actual matter in some instances, as well as projecting force fields to give the objects the illusion of substance. It's most likely food that is eaten in the holodeck is replicated matter and in this case, the water was also replicated matter and thus physically real, even after leaving the holodeck.
Corrected entry: As a non-corporeal entity enters and damages the ship's electronic systems, it causes the ship's engines to shut down. The ship, traveling at Warp 6, slows down and stops when the engines are shut down. However, in reality, ships in space only need the engines to get up to speed. Inertia will allow the ship to continue without the engines "pushing" the ship, because in space there is no friction to slow the ship when the engines shut down. The ship would continue on its current course at its current speed when the engines fail. Even if the warp field collapsed they wouldn't stop entirely.
Correction: The warp field bends space around the ship, it doesn't propel the ship forwards. If the warp field collapses, this bending of space would stop and as such the ship would stop moving.
Corrected entry: Data mimics the rest of the crew when Captain Sunad of the Zalkonians makes all the crew unable to breathe. Data is on his knees close to floor and is rising when "John Doe" heals the crew. Data is an android and should have just remained seated at his post.
Correction: Data is kneeling near Captain Picard and it is clear that Data is checking on Picard's well-being, because his captain has fallen to the floor and is gasping for air. Data is not struggling to breathe in any way, he's just looking at Picard so it is clear that he wasn't affected.
Corrected entry: They have already shown that anything that comes into contact with the time bubbles is immediately affected. The engine was running non-stop for 47 days, fruit aged, and Picard's hand aged too. Geordi creates a subspace isolation field that enables them to beam off the shuttle and onto the other ships for exploration. The issue is that the transporter beam would have stopped and never materialized as soon as it hit the time bubble due to the extremely slow passage of time. The warp core breach and disruptor fire were impacted, so it is highly likely that the transport beam would have been affected. Furthermore, they have not demonstrated the ability to modify the transporters to allow beaming over under such circumstances.
Correction: The subspace isolation field in conjunction with the phase discriminators were used to counteract this problem. Geordi and Data specifically mention how this would work. In addition, this same solution was used in the earlier episode Time's Arrow for a similar problem.
Corrected entry: After Picard, Guinan, Ro, and Keiko all beam back to the ship and have been transformed into children, their clothing hangs loosely on all four. Later, Dr. Crusher even states "But as far as we can tell, only their bodies were changed". Presumably they replicate a change of clothes that fit before the next scene. However, no one on the medical team addresses the fact that the now pre-pubescent Captain Picard has an adult-sized parthenogenetic implant (fake heart) that was mentioned in many previous episodes.
Correction: There is not enough information on the artificial heart itself to say with any certainty that it would have any adverse affects on a person were their body size to shrink. Since we don't know if it definitely would have any negative effects, we can't say it is a plot hole that it doesn't.
Corrected entry: When the colonists go into the cave to escape from the Crystalline Entity, you can see perfectly formed steps going down into the cave. The colonists were the first people there. Why would there already be perfectly formed steps in a cave?
Correction: There is nothing in the episode stating that the caverns had never been used before.
Corrected entry: Although they appear metallic, the Klingons' costumes are made of leather and squeak when the actors move.
Correction: How is this a mistake? Since we know nothing about the futuristic material used, there's no reason it couldn't look like metal but move like leather.
The Pegasus - S7-E12
Corrected entry: Apparently when it came to taking the ship into the asteroid, the entire bridge crew conveniently forgot that Galaxy class federation starships can separate. The obviously prudent course of action would be to separate the ship and send the star drive section (the narrower section) into the asteroid while the saucer section (the wider section) stood guard outside. The only reason this wasn't done was to provide drama for the rest of the episode when the Romulans collapsed the entrance to the chasm.
Correction: This is purely speculation. They may have needed the equipment, sensors, and weapons of the entire ship.
Corrected entry: Assuming he had smooth skin, Data has chest hair visible in a couple of shots.
Correction: It is well established in multiple episodes that Data has body hair. He was created to look as human as possible.
Corrected entry: Q talks about changing the moon's gravitational constant, and thus "altering its mass." Scientifically, changing gravitational attraction wouldn't affect its mass.
Correction: Q actually said "Change the gravitational constant of the UNIVERSE." In other words, change the law of gravity so that it functions differently and the moon's relative mass effectively changes, something members of the Q continuum are capable of.
Corrected entry: Even though living matter, especially complex living matter, cannot be created and taken out of the holodeck, the ship's replicators can create food which includes meat, salad, etc.
Correction: The ship's replicators are capable of synthesizing proteins, carbohydrates, and lipids to effectively create something that is very close to the taste, texture, and nutritional value of real food. Creating an actual living organism, however, is beyond the capabilities of replicators. Also, food created in the holodeck for consumption is replicated food, not holographic food.
Corrected entry: At the end of the episode, as Ro and Picard walk through the Bajoran camp, she states there is one condition to her continuing in Starfleet, at which point she puts on her Bajoran earring. In this episode and all other episodes she wears it on her left ear. However Bajorans wear the earring on the right ear, as stated throughout Deep Space Nine.
Correction: Since Ensign Ro is the first Bajoran we encounter in Star Trek history (in fact, Ro was originally supposed to be in DS9, but the role evolved into what became Major Kira), the mistake is in DS9, not TNG. Secondly, no one ever said Ro was a good Bajoran... there are plenty of Catholics who do the Sign of the Cross backwards.
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.
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.