Very often star streaks are seen backward. Ignoring tunneling (something that would have been hard to do in the late 80's) when moving faster than light speed stars would look streaked as depicted, but they would be blue as you approach the star and red after you pass. Sometimes this is depicted correctly but others it is seen the other way around.
As an Acting and later full Ensign, Wesley had all of the responsibilities and expectations of a commissioned officer which includes showing the proper respect to superior officers. Yet on several occasions, Wesley is heard being inappropriately informal, ie: Addressing superiors by their first and last names and rarely addressing any of them as sir.
When images of ships are moving through space, a rocket or engine noise is played. In the void of space, no sound would be heard. In space there would be no sound transmitted due to no medium through which sound (mechanical energy) to travel such as air, water, etc.
While there is a common belief that items in the Holodeck cannot exist off the Holodeck, there are several occasions where someone walks to another part of the ship carrying a trinket that was created in the Holodeck, or someone walks down the corridor while wet from Holodeck water. It appears that some objects (such as food) are actually replicated within the Holodeck, and thus are "real" (i.e. not holographic), depending on the needs of the program and its user.
The opening credits contain a compositing error. If you look at the sides of the ringed red planet, the stars on the left are moving an entirely different direction than the stars on the right.
The idea that Data cannot (more accurately, does not) use contractions in speech does not make sense. He is perfectly fluent in idiomatic English and various other languages (including French). He can therefore speak in whatever way he chooses. A running theme is that he wishes to be more human, so speaking common, colloquial language would be an easy start.
There are several episodes throughout the entire run of the series whereby the actors are clearly reading their lines off cue cards when speaking with a character via communicator or other transmission. It's noticeable that their eyes remain at a fixed point when delivering all their dialogue.
From Picard's ready room window, when the Enterprise is traveling at warp, sometimes the stars are seen streaking from left to right, other times from right to left. This would indicate the window as being located different areas of the ready room, which makes so sense.
While the bridge of the Enterprise is much more elegant than the bridge from Kirk's era, it is not built too well strategically. Whenever Picard asks for any information from Worf, he has to stand up and then turn to face him, or he has to get up, then walk up one of the side ramps to the stations above him to view that information. Or, if an enemy beams onto the bridge to kidnap Picard, Worf then has to either shoot and hopefully not hit a crew member below, or he has to jog down one of the side ramps to try and protect Picard. In addition to that, the ceiling of the bridge has a transparent bubble for an enemy to shoot through.
Despite the amount of pummeling that the Enterprise takes, whether it is from enemy phaser fire, disruptors, photon torpedoes, etc., no one ever has any loose items strewn about their quarters as one would expect. Even Picard's ready room is always immaculate.
Whenever the video signal is being lost, instead of pixelating, as a digital signal would, the picture shows analog "snow," which would be unheard of by the era.
Throughout the series, sometimes the comm badge emits a chirp when pressed, sometimes it does not.
Many times throughout the series, a stage light will be reflecting its light off a window, door, glass panel, etc.
During the first few seasons, the lighting on the bridge caused glare on the computer screens at the back of the bridge. To work around this, sheets of black cardboard were often put on LCARS screens, and this cardboard can be seen in many episodes. Changes in lighting in season 3 meant that the cardboard was phased out.
Apparently the Enterprise's shuttle bays are left unattended; there are many instances in TNG - as well as other ST series like Star Trek: Voyager (1995) - in which we hear of an "unathorized shuttle launch." It's extremely likely a ship with a military command structure would have personnel stationed in the shuttle bays at all times. However, it would make it more difficult for the writers to have someone get around the guards somehow, so it's an understandable omission.
The communicators don't serve as body cams, which exist today in 2021, so they would certainly exist 300 years from now.
Despite the Romulans having superior firepower with their plasma torpedoes seen from Kirk's era, that weapon is never seen nor even mentioned any time Picard encounters them.
It is claimed that Data can't use contractions (Can't, Isn't, Don't, etc) yet there are several instances throughout the series where he does. One of the first such examples is heard in Encounter at Farpoint (1987), where Data uses the word "Can't" while the Enterprise is being chased by Q's "ship".
Although the Federation and Starfleet use the metric system, there are numerous occasions when the ship's personnel refer to distances and capacities in obsolete units such as feet, miles, pounds, etcetera.
Throughout the series, Data learns various new aspects of humanity: friendship, figures of speech, bluffing, etc. He learns these as if they're new concepts to him. However, Data was activated in 2338, went to Starfleet Academy in 2341, graduated the Academy in 2345, and served on three Starfleet vessels before begin assigned to the USS Enterprise in 2364. It's hard to believe that in 26 years, Data didn't learn any of these concepts, but then was able to in the 7-8 years he served on the Enterprise.
Q repeatedly uses - and never correctly pronounces - the French expression "mon capitaine" when addressing Picard.
A key aspect of Data throughout the series is that he has no emotions. But on countless occasions, he can be seen expressing confusion, either by facial expression or in his verbal reactions. Also, in the first season, he occasionally will smile, or look pleased.