When Data enters his quarters, he walks up to the replicator and orders "feline supplement number 74" and a small, clear glass bowl materializes containing a portion of cat food. The quantity in the bowl greatly increases from the time it materializes to the time when Data sets it on the floor.
During Data's dance lesson in the holodeck, a towel can be seen on the wooden bar on the left side of the room. It's clearly visible in the overhead shots, and Dr. Crusher uses it and puts it back on the bar. When she leaves for sickbay, she takes it with her, but it immediately appears on the bar again.
When Data is petting Spot, right before O'Brien comes in, they switch cats. When looking at Data, the cat is bushy and has quite a bit of white under its chin. When the camera changes to behind him, the cat is a lot less bushy, and doesn't have much, if any, white hair under its chin.
Keiko wearing sticks in her hair is not inaccurate or "insulting". They are not chop sticks, they are Kanzashi hair sticks, which are a type of traditional Japanese hair ornaments. Some are fancy and made of gold and have jewels on them, while others are plain looking and made of wood, they can resemble chop sticks, the style depends upon the occasion and time of year. Regardless they have been in use for thousands of years by a number of cultures including ancient Egypt, Rome, Greece, China and Japan.
When Spot begins to eat from the cup of replicated cat food, a small insect inside the cup is disturbed and can be seen running across the food. In Star Trek: TNG, the food replicators are not programmed to materialize living tissue.
The appearance of the pads on the transporter platform is inconsistent. In some scenes, the border lines around each pad are more pronounced than they are during others. This is because the set was being redressed for use in Star Trek VI: The Undiscovered Country (1991). The thicker black border lines seen in that film were kept in for the rest of the series.
When Data is talking with Troi about marriage, an insect flies into the frame near Data's head as he says, "[...]because, I will not grow old."
When the ambassador is beaming over to the Romulan ship, Picard tells Worf to keep a transporter lock on her so she can be retrieved instantly if there's any trouble. It makes no sense to give this order to Worf, the tactical officer, rather than O'Brien, the transporter chief with whom Picard was already in contact.
When Beverly is teaching Data to tap dance, in the close-up shots of their feet, you can see the wooden floor move, showing it to be a padded portable modular click-together floor.
The night shift is seen on the bridge with the lights dimming, then the day shift with the lights becoming brighter. Interestingly, all of the other adventures occur only during the day shift.
There is no reason for the bridge lights to be dimmed during the night shift. Nights are dimmed in the evening to produce a more soothing environment and allow relaxation and sleep. Since all on-duty bridge officers must remain alert, it makes no sense to dim the lights.
While officiating at a wedding ceremony, Captain Picard says, "Since the days of the first wooden vessels, all shipmasters have had one happy privilege: that of uniting two people in the bonds of matrimony." This is false; whatever Starfleet's own fictional regulations may be, the captains of seagoing ships have no particular power to perform weddings and, at least for the US, the UK, and the former USSR, there is no evidence that they ever did. In fact the navies of the US, the UK, and various other countries specifically prohibit a commanding officer from performing marriage ceremonies. See also Balance of Terror (1966).
In Data's narration/letter to Commander Maddox, he mentions the Enterprise's new mission requires a change of course towards the Romulan Neutral Zone. Since such missions are usually classified or secret, he should not be discussing the mission in a letter to a non-crew member.
The character of Chief O'Brien is seen wearing lieutenant pips on his uniform, but it's a rank he does not hold as he is only a Chief of Operations. This happens several times throughout his time aboard the Enterprise.
Riker summons the captain by calling out, "Picard to the bridge." A lower ranking officer would always address a superior using an honorific. He could have said "Captain Picard to the bridge" or "Captain to the bridge." It is disrespectful in such a circumstance to use just the captain's last name.
After the ambassador's "death," Data explains in his log entry that he has noted that his human compatriots never fail to register grief at the loss of a comrade, and that is at such times he most misses the ability to share human emotion. To miss something means to regret the loss of something. As Data never experienced human emotion up to that point, he could not miss it.