Character mistake: When Matt is talking to his priest Father Lantom, he says "it was a nice sermon" instead of "it was a nice homily." In the Catholic church "homily" is the correct word; sermons are Protestant. (00:11:30)
Continuity mistake: A henchman places a briefcase up-side down on a bench, opens the locks and starts to open the briefcase with the locks above the lid latches which are flipped open toward the floor. Instantly the briefcase is right-side up with the locks down and the lid latches up. (00:26:11)
Continuity mistake: As they are fleeing the alien base, Supergirl and the others run towards the portal, Winn activates it and jumps through. In a following shot (after a 'friendly' alien helps them), he is standing behind Supergirl again. (00:32:00 - 00:32:35)
Factual error: There is a periodic table shown in the background of the lab, showing an accurate periodic table as of now. During 1945, at least 21 elements had not been discovered. (00:17:40)
Other mistake: John is promoted to Oberst Gruppenfeuhrer and assumes the uniform of such. He is soon after promoted to interim Reich Marshal. And then he is subsequently fully promoted to Reich Marshal. But he continues to wear the rank insignia of an Oberst Gruppenfeuhrer.
Suggested correction: John does eventually put on the insignia befitting a Reichsmarschall. It's safe to assume he didn't immediately put it on, either because it took a short time to produce new insignia and send it to him (they wouldn't stock insignia for a rank only a few people hold), or because he was still only acting Reichsmarschall.
Us vs. Them - S2-E2
Factual error: We see Liam's driver's license. It says he's from Rhode Island, but the license looks nothing like RI's. It's issued by the State Motor Vehicle Commission, but RI does not have an MVC (only New Jersey does). And the expiration date is always the same as the holder's birthday, but here we see it's 2 days after his birthday.
The Two Deaths of Jamie B. - S2-E4
Revealing mistake: When Naomi shows Kirsten the money, you can tell it's fake. Not only are the serial numbers all the same, you can read "this note is not real it is to be used for motion pictures."
Here There Be Dragons - S2-E11
Visible crew/equipment: When Prax leads the group to where Dr. Strickland took Mei, just as they reach the door the two green and yellow T-marks are visible on the floor. (00:02:30)
The Elephant in the Room - S1-E2
Continuity mistake: When Phil storms out during dinner and comes back in, he closes the door (we even hear it latch). When he proposes to Carol, the door remains closed until after she says yes, then in one shot it is open and stays open, with no one touching it.
Continuity mistake: When Luke and the others arrive at Gateway, there's a Smart car with its doors wide open. When they start walking and the shot changes, the doors are now hardly open. The doors keep changing position depending on the shot thereafter.
Continuity mistake: The quantity of super-steroid in Fukegao (the mad scientist) 's test tube changes continuously between shots. (00:11:00)