Episode #1.1 - S1-E1
Continuity mistake: As Ricky Gervais walks by the school where his nephew is, a "chubby little ginger" shouts "pedo" at him. He stops and holds one of the bars of the railing, third from the pillar. In the close-up, you can count more bars than that, showing he is in a different position altogether. (00:07:00)
Factual error: When Judy gets home and the key isn't working she looks at her iPhone and sees a long list of calls to Steve's mobile phone. This is wrong. IPhone just shows the name once per day and the number of calls in between brackets.
The Happiness of All Mankind - S1-E4
Continuity mistake: When Legasov proposes to use Bio-robots, Scherbina pours 2 glasses of vodka. One glass is about 80% full, another 60%. However, when glasses are placed on the table, they appear to be both filled to 80%. (00:39:27)
Continuity mistake: One moment Frankie takes out two shot glasses and a bottle and starts to pour. She pours maybe one-fourth of one glass. The very next shot, both glasses are filled. (00:03:35)
Plot hole: The captain's wife tells him that she's taking his daughter with her, not giving him a chance to even say goodbye. During the episode it turns out that she's angry because he missed lunch with them, and when the Metal Wu texts the boss, it also turns out that all this just happens at 1:11 AM! It does not make much sense: she waited over 12 hours to complain to her husband, and she's driving her kid in the middle of the night.
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."
Visible crew/equipment: When Michael Fay (1st Vampire ground zero) enters the cabin to escape the police, the film crew can be seen through the porch railing at the bottom right of the screen. (00:40:53 - 00:49:47)
Continuity mistake: When Nick's mom tells him to show Gypsy around she takes the cigarette away from her lips. A shot later it's back in.
Chaos Is the New Cocaine - S1-E3
Continuity mistake: Alex opens a bottle of wine. She takes the cork off the corkscrew with both hands and the bottle of wine spins from not showing the label to showing it. (00:37:55)
Continuity mistake: When Will Parry visits his grandparents, the grandmother is sitting on the sofa next to Will's grandfather. The grandmother's position keeps changing from sitting right next to him to being further apart.
Other mistake: The Zeds infect others through directly touching them. This rule extends to touching their clothes and backpacks, which makes no sense.
Factual error: When Jane is putting up the lost poster, the number to call is 1-800-saw-him. However, this is only a 6-digit phone number when it needs to be 7-digits (not including the 1-800 part).
Other mistake: The extent to which the classmates wrapped Miles in cellophane with his hands folded in front of his groin would make it difficult, if not impossible, for Miles to get out of the lake alive on his own. Miles did not drown... or suffer any injuries.
Continuity mistake: When Vincent is showing the delivery man his basement to show him the wooden cube isn't there, the delivery man's ID tag is positioned so that the front side is facing outwards. In the next shot, the ID tag has suddenly flipped around. (00:11:20)
Welcome to Hotel del Luna - S1-E1
Continuity mistake: When the woman on the street greets younger Chan-sung, her right hand is raised slightly when she says "Tell me the year and hour that you were born", but in the next shot, her right hand is suddenly in her basket. (00:13:58)
Revealing mistake: The ghosts often cast shadows.
Suggested correction: You can also hear their footsteps. But I'm not sure this is a mistake, just their own perception. Within the ghostly world they appear and act just like they would in life.
The ghosts are manifestations of energy, therefore they have no physical body to block light. Even if in the ghostly world, physics are physics and there should be no shadows.
Revealing mistake: The main character gets into a short wrestling match on the landing of a star well. The bad guy fires a shot over the main character's shoulder, which should have hit the wall he was leaning against, but no damage is shown on the wall. (00:05:40)
Continuity mistake: The Empire State Building is shown in white lights at 14.56 and 17.11 into the episode. The same night at 17.13, it is red, white, and blue.
Character mistake: In Season 5, Episode 4 ("The Heart Attack Boys"), Abishola twice gives Christina an injection without swabbing her skin with alcohol. Abishola is an RN and would never make that mistake.
Factual error: Season 1 Episode 1 Arrival - Doctor Jaax is a Lt. Col. in the US Army. Her BDU blouse has a US flag on the right sleeve. The time period of the show is 1988, as there is a doctor wearing a Dukakis T-shirt. The US Army didn't have a US flag on the BDU uniform until 2005. (00:11:20)