The Magician - S3-E4
Revealing mistake: When Oliver meets Thea at the Grind and Jot Cafe (Which looks unsurprisingly like CC Jitters) you can see the cups on the table are from CC Jitters. (00:20:50)
Character mistake: They are in the interrogation room questioning the fiancé of the scientist Natasha when the captain says, "the blood we found at the scene of your fiance's murder, it didn't belong to Bobby Cordero." The characters name was Benny Cordero, which the captain says correctly a few lines later. (00:33:25)
Death at Victoria Dock - S1-E4
Character mistake: Fisher refers to some characters as speaking "a Slavic language." The characters are Latvian anarchists. Lettish (aka Latvian) is a Baltic language, not Slavic.
Continuity mistake: When Bentley says how Maple Man can "gobble my knob", he briefly pauses between each word. When Theresa then plays back Bentley's confession, he says "gobble my knob" without pausing. (00:18:58)
Continuity mistake: When Buzz and Provenza race out of the squad room to follow Rusty in the frantic search for Stroh, Buzz is in shirt sleeves with no tie. However when they arrive at their destination Buzz is wearing a coat and tie.
Factual error: Noticed this watching season 2, and I assume it was also the case in season 1. The entrance of the Miramar Playa has flags from various countries flying. One of them is the Canadian flag with the maple leaf. Wrong flag. In that period the Canadian flag was the "Red Ensign", or "Dominion Flag." The maple leaf flag wasn't introduced until 1965.
Factual error: Chief Superintendent Hilton wears the ribbons of the General Service Medal (awarded for military service in a combat zone, most commonly in Northern Ireland) and the Police Long Service and Good Conduct Medal (awarded for twenty years' service), but not the Golden Jubilee Medal (awarded to all officers with five years' service in 2002), which he would also have. When he reappears in Series 4, he is wearing the Golden Jubilee Medal, Diamond Jubilee Medal (awarded in 2012) and Police Long Service and Good Conduct Medal, which would all be correct, but no longer has the General Service Medal.
Caleidoscope - S2-E4
Character mistake: The chess set in Dr. Pierce's home office is set up incorrectly. When the men are set up properly, each player has a light colored square in the right-hand corner. The board as it appears in the show is set up so that the light-colored square is in each player's left-hand corner. This "beginner" mistake puts the king and queen in the wrong positions.
Factual error: A plastic bag of items is found under the footboard. Plastic bags weren't invented in 1969. They first appeared in public in 1979.
Visible crew/equipment: When Rebecca Madsen jumps in her car to pursue the armored truck, as she does a U-turn and takes off at high speed, you can see all the people standing on the side watching the filming. Some even have their phone cameras out videoing it. (00:29:30)
Factual error: When Susan is sorting Ruth's things, she finds Gordon's iPhone wrapped in a newspaper from 1994 with her voice-mail to him and realizes he went back in time, where Ruth killed him and took the phone from his corpse. Even if the phone had been turned off, after 16 years idle, the battery would have died long ago, but we see the phone is on with her message waiting. (00:55:50 - 00:56:45)