Deliberate mistake: When the investigators are in the basement wine cellar, they turn on their flashlights to illuminate the pitch black area as they walk around. However there is a light source moving with the camera shining on them as they are casting a shadow behind them.
Deliberate mistake: On several occasions throughout both series/seasons, whenever a Terminator fires a weapon, they blink. This should not happen because that would be a human reaction - Terminators aren't human. They're made to look and act human, but SkyNet don't focus on the little things; such as emotions, breathing and reflexes. The three most notable instances of this are in Season 1, Episode 5: "Queen's Gambit", Season 2, Episode 21: "Adam Raised a Cain" and Season 2, Episode 22: "Born to Run". In "Queen's Gambit", Cameron blinks as she fires a machine gun at the T-888 in the back of the black van as he runs towards her, in "Adam Raised a Cain" the water delivery Terminator blinks as he shoots Savannah Weaver's nanny Debbie and as he shoots the security guard and Catherine Weaver in "Born to Run".
In Which We Meet Mr. Jones - S1-E7
Deliberate mistake: Dunham comments that she works for Homeland Security. While the FBI is under the jurisdiction of the Justice Department, she works as part of a special Homeland Security task force; so saying she works for DHS is an accurate statement.
Deliberate mistake: When filming the fake movie with Sophie playing the nun, many holes are made in the door, but when she slowly kneels to the ground, there isn't a hole behind her, so it seems like she's been miraculously saved. But as the scene continues, the nun starts to bleed and dies. Technically, they couldn't safely put a squib that close to her. (00:21:15)