
Factual error: Trust called Proxima B an "eyeball" planet, meaning according to him a planet that does not spin on its own axis and always has one side facing the sun. That is incorrect. If only one side faces the sun it means the rotation on its axis and its revolution around the sun are the same.Otherwise, the planet would seasonally have all of its surface facing the sun once every solar year.

Revealing mistake: Hannah sits in a conference room and pulls a landline phone towards her to dial her friend, asking about the will Owen left. The phone however isn't in any way connected, hence no call could have taken place.

Marathon-Thon-Thon-Thon-Thon - S1-E10
Continuity mistake: In the opening scene Dan has a newspaper in his left hand. When he walks away it has moved to his right hand.

Independent Dependents - S1-E7
Plot hole: When Helen goes into the air ducts, she simply removes the grates. They would not be loose like that, they would be firmly attached. When the team enters the basement they go through an ordinary door by picking an ordinary lock. A very high security installation like Axe Industries would not have such an insecure entry, even into the basement.

Visible crew/equipment: When Joel and Ellie cross the bridge, you can see crew on the left, in the overhead long shot. This show is an HBO Original, and the mistake has been edited out in the current streaming version. (00:14:35)

Plot hole: The Skrull base is inside an abandoned nuclear power plant with enough radioactivity to force any human (like, say, Nick Fury) to constantly pop iodine pills to fight the symptoms of a poisoning that would kill them in less than half an hour. Despite that, Skrulls also detain prisoners, for years in some cases, in rudimentary shackles without any sort of shield or protection against the radiation.
Suggested correction: Iodine pills don't fight the symptoms of radiation poisoning; they prevent the body from absorbing radioactive iodine. It does not protect from exposure to radiation; it won't save you from it. Secondly, it's all an act by Gi'Ah posing as Fury anyway. Thirdly, they are in the reactor control room where Gravik says the radiation is higher. The prisoners are in a low radiation room, which could be extra shielded from radiation. It could also be that the prisoners are fed iodine to block radioactive iodine.
We can make up if we want that there's a special, super-secret anti-radiation serum and/or super-effective shielding, helping humans even during an exposure that lasts years (a decade in the case of Rhodey!), but there has to be something in the actual visuals that remotely hints at it. It's hard to headcanon that the dingy area of the plant where they are racked together, strapped to bed nets behind tarps, can be "low radiation", or that they are given anything to counter it. In particular, in the ending, the rescued people leisurely walk around the plant with zero radiation protection, even casually in the open yard where "Fury's" Geiger counter was going mad earlier. And the radiation was not something induced by the Skrulls that just ended when the baddie died. Not only is there no techno-babble justification (one could argue it's simply a pedantic detail not unlike the lack of hair growth or muscle atrophy), there's a direct flagrant contradiction in how the environment of the location - which is the only reason why they picked that site as a base - is deadly to humans only to a dramatic degree only when it's convenient.

La Rochelle - S1-E1
Continuity mistake: S1E1 La Rochelle. Kate is the new First Officer of the cruise ship, later she introduces herself as First Mate - a much lower rank.

Factual error: The Royal Navy sailors are shown with just 'HMS' on their cap tallies. This was only seen in the Second World War for security reasons. In peacetime, the tally bears the name of the ship or shore establishment to which the rating is assigned.

I Become Supreme Lord of the Bathroom - S1-E2
Continuity mistake: When Percy comes round at the beginning of Episode 2, he is wearing a different shirt than the one he was last seen wearing at the end of Episode 1, in which he lost consciousness. (00:00:01 - 00:01:00)

Factual error: Season 1, Episode 10: Lee is trying to get out of the underworld on his capsule with the rescued others. They all get in, and he closes the top hatch by turning the hatch wheel clockwise. Trying to take off, a wire becomes disconnected outside, and Lee has to go out again. He turns the hatch wheel clockwise, and this time it opens instead of closing. If it was clockwise for close, it should be counterclockwise to open.

Continuity mistake: As Jizzlord says, "so really, this is a good thing", he's holding the posters upright, facing Jen. Cut to a shot from the side and he's holding them horizontally, facing upwards.

Other mistake: Ms. Herfkens is relating the final moments when the aircraft she was in crashed in the jungle. She is sitting to the left of her fiancée. After the crash, she states what happened after the crash. She states that she looked to her left and saw her fiancée still strapped to the seat. But if she was sitting to his left, as shown in the scene, she would have had to look to her right. Given that she is recounting an actual event, the scene showing her sitting to his left is wrong.

Factual error: Karl Silberbauer, who headed the raid on the secret annexe, wears the insignia of an SS-Sturmbannführer (major). He was actually an SS-Hauptscharführer (senior sergeant). His superior is addressed in German as "Herr Major" (not an SS rank and not how an SS officer would be addressed), but wears the insignia of an SS-Standartenführer (colonel).

Continuity mistake: A nurse's driver's side rear door is kicked several times by a man. The nurse gets out of the car and tells the man, "You dented my car." When she gets back in the car to drive off, there is no dent in the door. (00:26:04)

Such Inward Secret Creatures - S1-E5
Audio problem: After Paul and Amy make love, Paul puts his pants back on and stands back up. He says, "What?" There is a strangely timed "what" right after that. It's an exact replica of the first one. There is an audio overlap from the other angle here.

Factual error: Captain Carrillo is target shooting with a hand weapon. She empties the magazine, but she catches the magazine and hands it to her ranger partner instead of letting the empty magazine fall to the ground. As a trained military officer and training with special operators, catching an empty magazine is an extremely bad habit and practice, which is a big no-no. (00:40:23)

Thank You for Not Driving - S1-E3
Other mistake: The scene in which Tayo's wife gets into her car to leave their home is filled with mistakes. First off, it seems to take her much too long for the short driveway to reach the curb. Then, when she is blocked by the perp's van, she rams one of the perps twice and then backs away. As she is backing away, the perp's body flies forward, smashing into the van. She then rams the van twice with great force, and yet her car has minimal damage, only to the corner of the car.

Factual error: In the beginning when the girl's cell phone goes crazy, she says that she should have gotten a Razr. Razr didn't come out until 2004, and this scene takes place in 2002. (00:01:09)

Other mistake: Juliette and two cohorts need to go from Level 22 down to Level 125. Using the normal circular stairs, it would take several hours if not days to get there. They enter a garbage chute and climb down using a vertical ladder on the side - a very clumsy and slow descent. Yet, they make it in what appears to be a very short time.