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)
While You Were Sleeping - S1-E2
Other mistake: When Sherlock was slapped, he later says he was surprised because the girl who slapped him was a "lefty." Then he rubs his right cheek and says it's leathery from all the slaps he's gotten, while the left is soft. This is backwards - anyone hitting him with their right hand would hit his left cheek.
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.
Factual error: The depiction of the hanging is completely inaccurate. By the 1890s, the gallows at Newgate were housed in a room within the prison, not in the courtyard. Prisoners' wrists and legs were pinioned before they were hanged, not left free as depicted. The long drop was used in Britain from the 1870s, not the short drop depicted. The large knot depicted was not used in Britain (although it was in America) ; a simple sliding loop was actually used, and this was positioned under the jawbone at the side of the neck, not at the back of the neck, which would quite probably not have broken the neck, leaving the person to strangle to death.
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.
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.
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)
Turtles All the Way Down - S1-E13
Other mistake: When Dr. Lee and Dr. Evans are arguing just after Red Reality Michael and Green Reality Michael converse, Michael turns around and says, "Shut up!" but the closed captioning says, "Enough!"