The Funeral - S1-E3
Deliberate mistake: Though it serves the drama and black comedy of the scene, it's highly unlikely a modern, let alone future broadcasting network would neglect the importance of delay (usually around 30 seconds) between recording and "live" broadcast. Specifically, to prevent unexpected gruesome incidents like the exploding head seen in this episode from making it to air.
Factual error: The communication delay for earth-moon communications is around 2.6 seconds. Yet Emma's daughter simply dials a cell phone and is connected to her Mom in space past the moon in less than that. The communication has no time lag. (00:18:45)
Remembrance - S1-E1
Character mistake: Picard and Data are playing poker at the beginning of the episode. Before the actual faces of the players are revealed, we are shown close-ups of their hands as they put down the chips for the bet. When Picard says his first line in the series it's "See", and it is literally true, because in the previous shot, Data put his 5 cards down on the table and face up. (00:00:50)
Factual error: In Massachusetts, Atticus, Leti and George are attacked by the local sheriff and his deputies, who are patrolling the county roads. In Massachusetts, sheriffs run the county jails, but they do not (and did not) handle routine policing, which is the responsibility only of municipal police departments.
Factual error: Paul tells Shea that he has a condition called Sporadic Fatal Insomnia. Later, Shea finds a fork in a bag labelled Abby. Paul explains that Abby is his daughter, and he wants to test her DNA, because she has a 50-50 chance of contracting the disease. This is false. Sporadic Fatal Insomnia is not hereditary. Fatal Familial Insomnia is genetic and is the inherited form.
Continuity mistake: The three friends finally meet in person after all the texting. Karube brings a cigarette to his mouth when Chota asks "Where to?", but in the next shot his hand is lower. (00:07:15)
Revealing mistake: As Ed slowly backs away from the camera (from the foreground to the background), the towering Scrapper robot moves slowly forward (from the background to the foreground) to take Ed's place facing the camera. The lumbering robot is plainly casting a long shadow across this night-time shot towards the right side of the screen. But, as Ed and the giant robot pass one another, the CGI robot's shadow never falls across Ed at all. Throughout the shot, Ed remains uniformly illuminated, revealing that the digital-effects artist neglected to add the shadow passing over Ed. (00:38:55 - 00:39:19)