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.
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.
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.
Character mistake: Issei meets with a journalist to tell her that he has decided to end his participation in the contest to win the estate. She hands him her business card, which he immediately puts in his pocket. This completely violates the Japanese traditional business card ritual, which includes looking carefully at the card (a sign of respect).
Other mistake: Agent Maddy is approaching the scene of a downtown explosion and fire. She has her FBI badge showing on her belt. As she approaches the police line she rummages in her jacket and looks down at where the badge is, but reaches into the adjacent pocket to pull out another badge in a wallet. Why not identify herself with the badge on her belt?
Factual error: There are two Marines posted at Marine 1, the Presidential helicopter. Their pants are ridiculously long and not the image of Marines with well-fitted uniforms. Their position of attention is wrong, with the hands clenched and placed on their thighs as opposed to being fixed at the sides. Worse, when they saluted with their right hands were above the service cap brim, instead of the fingertips touching the bottom of the brim.
Deliberate mistake: Identical twins Beverly and Elliot have the exact same beauty mark on their faces, roughly on the smile line directly across from the right nostril. Identical twins don't share features like that.
Continuity mistake: When talking on the roof at the start, the guy in green has his hands in his apron pockets when being told the security cameras are off. Cut to a wider shot and his arms are folded instead.
Protect Her - S1-E1
Factual error: When Jules and Hannah are talking about the financial shenanigans at Owen's firm, Jules says that the owner of Owen's company had fraudulently driven up the price of the stock before the IPO. This is impossible. The IPO is the first time the shares become tradeable.
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.
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.