Continuity mistake: When we see Alfred in the Batcave, he had 4 neat piles of playing cards and turns over an Ace. When Batman arrives, there's a shot of the cards, but the Ace is no longer turned over. When he makes the comment about seeing another playing card, the shot changes and there's now 5 piles but not in a neat pile.
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.
Factual error: In s01e08, "Q and A", when Yelena is playing chess, we see her use the touchscreen which represents the chessboard. However, the bottom left corner is a white square when it should be a dark square. The board is orientated incorrectly, even though the board on the floor isn't.