Character mistake: Bruce and Alfred play a short game of chess, after which Alfred remarks, "I don't know why we bother playing this game - you never lose." The problem is, Alfred moved into a position wherein Bruce could take his king; this is called "moving into check" and is not allowed in chess.
Character mistake: In s01e08, "Q and A", Yelena moves her bishop to B6. The announcer says that's going to win it (the game gets interrupted at this point). However, that move does not put the computer (her opponent) into checkmate, it doesn't even put it into check. Her bishop is on a dark square and the computer's king is on a white square.
Character mistake: When Shikata disappears after she loses her powers, Peter drops the sword and he, Mary Jane and two pedestrians behind them are shown looking in the direction that Shikata was just standing in. A third pedestrian, standing to the left of the shot, can be seen but is looking in a completely different direction to the others. (00:18:37)
The Strike - S2-E4
Character mistake: The boy says he's on McConnell Street, but the street sign says it's McConnell Avenue. (00:16:03)
Character mistake: Joe called Cruz a Sergeant First Class (E-7), which is a US Army rank. It was established in Season 1 that Cruz is in the US Marine Corps, so she would be a Gunnery Sergeant (E-7). (00:19:48)
Behold... The Inhumans - S1-E1
Character mistake: When Hunter One and his men are trying to sneak up on Gorgon in the jungle from behind them, Gorgon chuckles and says, "Never approach downwind, dumbass." He then proceeds to counterattack them. This implies he smelled them coming with the wind. However, this is backwards, as they would need to approach from upwind for their scent to carry to Gorgon. Approaching from downwind would be correct. So he should have said, "Never approach from upwind, dumbass." (00:33:35)