The End Is the Beginning - S1-E3
Continuity mistake: In the last phase of her conversation with Picard, when she sends him away, Raffi changes position between shots, so her pigtail is on/off her knee. (00:15:30)
The End Is the Beginning - S1-E3
Continuity mistake: When Commodore Oh appears in an odd Men In Black cosplay to Dr. Agnes Jurati in Okinawa, Agnes removes her earplugs and lowers her arms, but in the next shot she is still shown with her right hand raised. (00:16:15)
The End Is the Beginning - S1-E3
Continuity mistake: When the hologram is removing the shrapnel from Cristobal 'Chris' Rios' shoulder, he holds the tool at an entirely different angle depending from the shot. (00:20:30)
The End Is the Beginning - S1-E3
Continuity mistake: When Raffi figures out Maddox's location through her drunken handwaving, she is holding the jar by the neck at different heights between shots. (00:22:20)
The End Is the Beginning - S1-E3
Continuity mistake: Zhaban put in a bag some delicacies for Picard. When he mentions "Madame Arnaud's terrine d'oie" his hand is holding the jar in the first shot, during which you can see his hand turn to hold one of the bag's handles. But in the reverse shot he is still holding the jar, and puts it in the bag. (00:27:10)
The End Is the Beginning - S1-E3
Continuity mistake: After the action scene at Chateau Picard, we see Soji talking with Ramdha. Ramdha is placing the romulan tarots in configurations not coherent between shots; for instance when Soji says "the pixmit, how do they work?" there's a space she'd wedge the card in, but in the overhead close-up she is putting it to the side of two cards. (00:31:00)
The End Is the Beginning - S1-E3
Continuity mistake: In her quarters Soji tells Narek "If you had asked me 5 minutes" etc.; her mundane-looking special necklace pops out of her shirt's collar on her right. After a brief reverse shot when she does move her neck, her necklace is all the way in front of her collar on the left. (00:36:05)
Character mistake: Dr. Agnes Jurati, one of the Federation leading scientists, is bored during the hyperspace travel, and so she chats a bit with the captain. While she gives her quirky speech, she casually mentions that "there are over 3 billion stars in our galaxy." She's not wrong, technically, but the number of stars in our galaxy is estimated between 100 and 400 billion. She is way off. (00:08:40)
Suggested correction: Two things. You state that she's not wrong, which she isn't. The fact that she chose an odd turn of phrase doesn't make it a mistake. Plus, you reference the number of 'galaxies' in our galaxy but I am guessing this is just a typo.
Plot hole: Mr. Vup is a Beta Annari, and they can, as it is stated (and for comedy purpose stated again) literally "smell" lies. However, Raffi gives Rios a unique concoction that camouflages lies, and it is made of drugs (beta blockers, anxiolytics, benzos). At least two things don't make sense here. First, Picard gets no shot and his whole flamboyant performance is one big lie from beginning to end, but he is not sniffed out - you'd also assume they could easily tell he has both eyes, since they have various detectors. Second, when the substances kick in as Rios is forced to lie openly, even us the audience, as olfactory-impaired as we are, can see he is getting high as a kite from him making a funny face; a species that can detect subtle changes in a metabolism over a simple lie, surely would detect when someone has such a dramatic alteration in front of their eyes - and see that as a telltale sign of something fishy going on.