Character mistake: When Ginnifer Goodwin leaves the supermarket to go the parking garage, the green sign says 1 hour parking Customer's Only. It should say Customers Only, without the apostrophe.
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.
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: When Harlan Edwards collapses and Dr. Arcane tells them to clear his airway, one of the other doctors addresses her as "Dr. Cane."
Character mistake: When meeting with the network exec in the bar in the opening scene, Bradley orders bourbon on the rocks. When her drink comes it is without ice.
Character mistake: The date on Louise's plane ticket to London is listed as "Thursday 16 February 2017", but the month "February" is missing the U, misspelling it as "Febrary." (00:15:56)
Character mistake: At the police station when Damien Asante is reading the information on the computer about the hit and run, he says "On February 25, 2015 Christy Markus and her son were struck by a car. The driver kept going." If you look at the computer screen, the date of incident says February 25, 2018. (00:30:00)