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 Star Trek: The Next Generation: Encounter at Farpoint, where Data uses the word "Can't" while the Enterprise is being chased by Q's "ship".
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.
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.
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.
Items in the holodeck cannot exist off the holodeck. However in the first episode Wesley falls in the river where Riker and Data met, and was still wet going down the halls of the Enterprise. Also in "The Big Goodbye" which is the first time we see Picard playing Dixon Hill, he had gotten a kiss, and when he left the holodeck to change into a suit, he was walking down the hall of the Enterprise with lipstick on his face.
In eight appearances as Q, John de Lancie repeatedly used -- and never correctly pronounced -- the French expression "mon capitaine" when addressing Picard.
Many times throughout the series, a stage light will be reflecting its light off a window, door, glass panel, etc.
Repeatedly, the crew refers to the "ship's computer," as if the ship has only one computer aboard. This is likely because when the original series was made computers were large and bulky and the mainframe model of computing was in use.
More likely, the Enterprise would have had millions of computers networked, with central file servers.