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".
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.
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.
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.
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 - 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.