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.
Plot hole: Dr. Adam Soong is initially presented as a discredited scientist, banned from the scientific community; he gets debarred and his funding revoked. And it's not an internal matter; he is publicly exposed for it. His daughter in episode 6 even finds out this information on Google. Several news articles call him "mad scientist" and such. However, this same person at the same time throughout the rest of the season has every bit of pull and influence, not just through undercover channels, but is treated with the utmost honor and deference by the NASA PR people at public events.
Plot hole: In this episode, the 'Watcher' displays the power to possess/mind control people at will, jumping from body to body to set up the meeting with Picard. Forgetting the fact that this usage of power for such a menial task is actually detrimental to what she wants to do (it leaves more evidence, by unnecessarily messing up with minds just to tell the guy to go from A to B), this would have been super-useful for the rest of the season, but she just never ever uses it again, not even a nerfed version of it.
Other mistake: The show is supposed to take place in the same timeline as Next Generation, but the presence of a young Guinan in 2024 who has never met Picard before contradicts the TNG Season 5 finale, Time's Arrow, where Picard met Guinan (looking like Whoopi Goldberg, who also appears here) in the XIX century. When asked about this contradiction, show writer Terry Matalas said that the Federation actually never happened since the 'future' was changed by Q's actions, so Picard never traveled back in TNG meeting her. However, that is never referenced in the actual episode or the season finale, and would create other paradoxes, especially since in the very same episode there is a reference to Star Trek IV. The same punk from back then rubs nervously his neck remembering Spock's nerve pinch; the TOS cast would have not visited XX century Earth in the way shown in the movie if the Federation never came into existence.
Plot hole: Adam Soong tosses the phaser before it explodes. The explosion is an underwhelming bang in the air that does not hurt anyone. Soong is a chubby old dude, with no access to teleport technology or any tech, no allies left in the area, inside Picard's home in France, and all he did was run out of the door, unarmed. All Rios (or anyone else, really, arguably even Picard-bot) has to do to catch him is comfortably jogged through the corridor, but somehow since he went out of the camera view and the plot doesn't want them to look for him, he ceased to exist, and he can stir trouble later after he gets (somehow) back to California.
Character mistake: Seven of Nine asks Tallinn to lay down suppressing fire, so they "split up." She explains "We have twice as many chances to get to the ship if we divide and conquer." 'Divide and conquer' (Divide et impera, for the Romans) means to keep the opposition weak by favouring the conflict between their different factions. It is not used to describe splitting up your own forces to have better chances to make it out of a trap alive.
Other mistake: When Tallinn dies, not only she is killed through a neurotoxin made in 2024 by a human, for humans, through a patch that the killer carelessly touches himself (we have to be generous and assume he developed some antidote to it) and somehow they can't cure with their advanced tech, but her eyes become bloodshot with visible red blood vessels. Romulan blood is green.
