Continuity mistake: When Riggs is hanging from the tree, his hands are tied behind his back. When he falls and lands on his back, his hands are still behind his back. In the next shot of him, his hands are now tied in front of his body, even though he didn't have time to do anything to get his hands there.
Left Behind - S1-E9
Factual error: Ray, Sara, and Kendra are Left Behind in 1958. However, 10 weeks later Sara and Kendra are playing the board game "The Game of Life." The modern version, as seen, didn't come out until 1960 (previously it was called The Checkered Game of Life and doesn't resemble the modern version).
Continuity mistake: When James is taking pictures from a car as men are entering a hotel, in the shots of him head on he is using a Canon EOS camera with a 70-200mm white lens. In shots from the side he is using a Nikon camera with an older Nikon lens. (00:27:35)
Plot hole: They only take out the shrapnel from the gut shot - they never deal with the shrapnel in his shoulder at all. (00:04:25 - 00:05:00)
Seeds + Permafrost + Feather - S3-E17
Factual error: Bozer and for that matter I would assume all of the team went to a spycraft academy. A basic subject in any of such academies is surveillance, including proper techniques to follow and stake out someone. Bozer being so inept at tracking MacGyver's father is inconceivable given their training.
Factual error: When the trucker reaches into the bag that's suppose to have his cash with his "bonus", he's bitten by a venomous snake. However, we see the snake and it's a non-venomous king snake.
The King of the Delta Blues - S2-E6
Character mistake: "Are you going to send in the calvary?" is said twice. It should be "cavalry". (00:09:40)
Other mistake: Issac is on the phone with his boss and he is holding his phone upside down, noticeable because his otter box case is upside down. (00:14:41)
Continuity mistake: When Dmitri and the doctor first try to get Vanessa to feed on human blood, the blood bag keeps switching from full to half-empty from one shot to the next. (00:16:05)
Continuity mistake: Harry drives away from a crime scene and his rear numberplate is visible as D703 VAN. We then see Harry in his car from the front, and the numberplate has changed to E160 HLT. (00:21:18 - 00:22:31)
Continuity mistake: When Will tackles Andrew, his shirt is lifted up exposing a blue undershirt that's tucked in, which is also behind his gun. There's two shots of this, but in the next shot, right before Will punches Andrew, we don't see the undershirt.
Episode #1.4 - S1-E4
Other mistake: A major plot point is the rivalry between the Wehrmacht and the SS, who refuse to work together. After the explosion at the cemetery, it is the Wehrmacht who are explicitly described as rounding up and interning suspected resistance fighters, with the SS not being party to it. However, when Archer visits the internment camp, most of the staff are wearing SS uniform.