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.
Character mistake: Artie says he'll recite pi to 100 decimal places. After the first few numbers, it skips to him finishing up by saying "46375." The last 5 digits would be "70679." In fact, the sequence "46375" doesn't occur until the 95,517th position.
The Penguin: Please, now that all pretenses are off, call me Penguin - a flightless bird, but one with style.