Factual errors: Any time Data plays the part of Sherlock Holmes he's shown as wearing a deerstalker cap and smoking a goose necked pipe. Holmes only wore a soft cloth cap (drawn as a deerstalker though never said to be one by Holmes creator Arthur Conan Doyle) when in the country, not in the city. Also Holmes never once smoked a goose necked pipe. That was an alteration made by an actor who found that it was the only pipe he could hold between his teeth and still be able to say his lines clearly. Finally, Sherlock Holmes never once spoke the words "It's elementary, my dear Watson." That was something else added in a script but never once written by Doyle. An android would get these details right.
Continuity: In some episodes, every person in the turbo lift has to state their destination before it moves, while in other episodes, only one person needs to state a destination.
Continuity: In some episodes, crew members need to touch their communicator and state who or where they are and the person or place they wish to communicate with ("Riker to Picard", "Bridge to Engineering", etc.), while in other episodes, one only needs to state the information without touching their communicator, possibly using the ship's built-in communication system, but if that were the case, why use the communicators at all when they can just use the ship's built-in system. Additionally, the communicator always knows the correct person to contact even if there's more than one matching name (there's 1000+ people on the Enterprise, most of them crew members, it is unlikely that none of them share the same name).
Errors made by characters (possibly deliberate errors by the filmmakers): It is established that Data (Brent Spiner) can't use contractions (Can't, Isn't, Don't, etc) yet there are several instances throughout the series where he does. One such example is heard in the pilot episode, where Data uses the word "Can't" while the Enterprise is being chased by Q's "ship".