Chapter 9: The Marshal - S2-E1
Character mistake: Cobb Vanth says that he's never seen anything but the dragon's head and had no idea about the size of the dragon...but at the beginning of the episode we could see the beast pass through town (conveniently right in the main road without damaging anything else) and its enormous lengthy body was perfectly noticeable. (00:31:20)
Character mistake: Batwoman does everything she can to rescue the elevator falling down after the villain detonates the bomb, but does nothing for the second one Tommy blows up to make her lose her footing when she is vulnerable. There is no message from Luke telling her anything, and she is not shown having a HUD with thermal readings or something like that, so there is simply no reason why she would risk everything to stop one elevator and completely ignore the fate of the next one which could be full of civilians as well.
Character mistake: When Harlan Edwards collapses and Dr. Arcane tells them to clear his airway, one of the other doctors addresses her as "Dr. Cane."
When in Rome - S1-E3
Character mistake: Nobody in this scene knows how to dance the tango that is playing. (00:16:35)
Character mistake: Shaun says that the underground facility is "built like a bunker", but the single missile they dodged just created a huge hole a moment earlier. (00:11:40)