Continuity mistake: In the Batcave, Kate holds the childhood picture with her sister by the sides in the first shot, by the bottom in the close-up. (00:08:40)
Continuity mistake: When Kate is writing the diary entry for Bruce, in the close-up the written part is taking barely half of the page, while in the overhead shot it is taking way more - the last lines being presumably actually scribbled by Ruby Rose have a much wider spacing. (00:40:20)
Continuity mistake: The hair on Sophie's cheek as she stares at Batwoman after landing on top of her and is in kissing range is parted in locks that move and differ across shots in noticeable way (for instance when Kate pulls the gag off her mouth there's a very thin one curling to her lips, but it's a thick lock right after). (00:35:00)
Continuity mistake: When Alice is talking to Jacob Kane with Sophie on the plank, she says "Can your little birdie fly?", holding the radio with all fingers straight and on the side of the device. Cut to "For if she moves an inch..." and her fingers are curled, with the pinkie at the bottom. (00:33:15)
Revealing mistake: In the Batcave, the protagonist looks at the newspaper clipping about the death of her mother. The rightmost column has a headline "Children traumatized after ride of terror", and a sign ("by Lucy Moguire"). And yet, it begins like this; "With the population of the planet ever increasing, so does the demand on Earth's resources." It's a lazy copy and paste by the prop department, putting some global warming report as filler text. (00:27:40)
Continuity mistake: The quantity of blood behind Batman's dummy changes; in the overhead shoot of the crowd it runs horizontally along the step for an area 3-4 inches wide, while in the subsequent close-up there's no such trim. (00:12:00)
Who Are You? - S1-E4
Factual error: At the end of the episode, Batwoman shoots her Bat-rope at the villain and yanks her off the building she made her escape to. Magpie is not a superhuman invulnerable being; assuming the grapple didn't puncture anything in her flesh and just hooked to some part of the costume like her backpack, she still is pulled with force from well above the 5th floor of a building to the end of the street, back first into a fountain, with a water level so shallow that she backpaddles into it as soon as she lands without being submerged. She does not experience a single moment of shock or stun from an uncontrolled fall of that magnitude. (00:33:00)
Plot hole: Batwoman sees live on TV the explosion that sends the truck into the river, while she is in the clinic of her stepsister on top of a building. She needs to go get her bike and drive to site. Fox says that she is "5 minutes out" when the truck is shown going completely underwater, with Alice inside who has been unconscious all the time. Even assuming that she makes it in less time than that, she still has to leave her bike somewhere and access the river from a somewhat secluded spot and swim to the spot, because the bridge is crawling with cops. There is no way that Batwoman would be able to save her and Alice would be totally fine, needing simply an aqualung to get back to her senses (and somehow survive an explosion that nearly kills Batwoman with her padded 10 million dollars suit).
Continuity mistake: In her diary entry at the end of the episode, Kate puts the pen down to fetch the file. She puts the pen by the diary at first, but in the next shot the pen is on it, in the middle of the page. (00:40:20)
Who Are You? - S1-E4
Plot hole: The whole "booby trapped frozen room" does not make any sense. Let's try to ignore the fact that it is entire speculation and a huge leap of faith on Mr. Fox's part that the trap works that way or that it's the only one, and that it wouldn't make sense (especially for a thief) to have something so erratic rigged to destroy your whole expensive equipment and jewelry stash - at most something directed at the door or a small charge to destroy your HD to cover your tracks. Still, it is supposed to react to any minuscule variation of temperature in the room and is triggered by a small sneeze, but Batwoman opens the door normally, and that alone would create a variation in temperature much bigger than her breath's (just think of what happens when someone does that in real life!). (00:21:40)
Revealing mistake: During the sequence with a crowd in front of the Gazette looking at the dummy on the stairs, there is a shot with the camera panning up (right after Kate arrives). Look in the crowd on the left side; a kid is following the movement of the camera on the crane instead of looking at the 'crime scene'. (00:11:50)
Who Are You? - S1-E4
Plot hole: Fox is alerted by the alarms that there is an intruder in the house, exactly in Wayne's office. He has to make his way up from the Batcave to the main residence, enter by the door and then he gets knocked out by Magpie. As it turns out at the end of the episode, she had the place perfectly scouted (and it was shown anyway that the intruder was already in the office when the alarm rang), so there's no reason at all why she'd still be there to assault him, considering the necklace was not guarded by some bulletproof crystal or anything of the sort and if she planned such a bold theft in broad daylight she surely wouldn't take her time to get what she came for. (00:15:25)