Other mistake: When the 3 go to Ivy Town, "Disco Inferno" is playing. However, this is 1975 and the song was recorded and released in 1976.
Character mistake: After Ray Palmer says he's an Eagle Scout, he says to Leonard Snart, "be helpful to others, scout's motto." The Boy Scout's motto (including Eagle Scouts) is "Be Prepared." "To help others at all times" is part of the Scout's oath, but that's different than its motto and not something an Eagle Scout would mix up.
Suggested correction: It was a joke, that applied to the situation. Wasn't meant to be literal.
There is zero indication of any joke, implied or otherwise, especially since he could have correctly said "oath."
Factual error: At the beginning of the episode, when Snart stole the female soldier's keycard in 1986, she is wearing the U.S. Army's Combat Infantryman Badge (CIB), a combat award for infantry and Special Forces soldiers. It is not possible for a woman to have a CIB in 1986 because infantry and Special Forces were closed to women at the time.
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).
Left Behind - S1-E9
Plot hole: It's revealed that Chronos is Mick from the future after he was abandoned by Snart. However, in the episode White Knights (episode 4) the earlier version was able to sneak up on Chronos and surprise him with an attack from behind. If that was a future version of Mick then surely he would have remembered the attack coming and could not have been so easily caught off guard.
Continuity mistake: In the flashback to 1920, Kendra flips the calendar to October with a snowy mountain side picture. Later when she's talking to Carter about leaving, the calendar is now on September with a trail through the grass picture. It goes back to October when their son comes in.
Last Refuge - S1-E12
Factual error: In 1993, when Jefferson's father James comes to the hospital to see his son after he is born before he deploys to Somalia, he is wearing his rank upside down on his uniform collar, and he has no rank on his headgear. Also, he wore his headgear while he was indoors for the duration of the episode. Headgear is not authorized to be worn while indoors except in certain cases, none of which would have applied while he was in the hospital visiting his son. (00:18:15)
Continuity mistake: In the scene aboard the Waverider when the crew discusses their discovery of the Oculus, the first two times you see Sara sitting on the Captain's chair from the back she's leaning forward with her elbows on her knees. When you see her from the front, though, she's sitting with her legs crossed and her right hand on the control panel, and in a following shot you can see her actually change her position and lean forward. (00:23:45)
Out of Time - S2-E1
Other mistake: Sara tells Jefferson that the professor has a one hell of a right hook (referring to Martin knocking Einstein out). Jefferson replies, "seriously, Gray punched him out?" The subtitles mistakenly say "Ray."
Continuity mistake: While all the heroes are in front of the aliens ready to fight, the sun on the floor changes between shots.
Character mistake: When the Legends are talking about Chicago in the 20's, during prohibition, Nate says there was "illegal drinking." But prohibition didn't make drinking alcohol illegal. It was only the production, importation, transportation, or sale of alcohol that was illegal. As a historian, Nate would know this and not make that statement.
Suggested correction: The term "Illegal drinking" is actually in reference to people going to locations that were selling alcohol illegally to drink it, hence it is illegal drinking.
That still would not be illegal drinking.
Factual error: The astronaut in 1970 says "run any test you like I just want to be on that shuttle". The shuttle wasn't in operation until 1981, the mission he's referring to is the Apollo 13 rocket. (00:39:50)
Factual error: When Ray is on the moon, he's getting the spear from the flag that was planted in the Sea of Tranquility, from the Apollo 11 mission. When we see the flag, it's still standing up. However, Buzz Aldrin saw, and reported, that the flag fell over from the rocket's exhaust as they left the moon, and no one from Apollo 12 put it back up.
Continuity mistake: When Ray talks to Alan, there are red binders behind Alan. The one on the end switches from being tilted and leaning up against the other book to standing straight up, depending on the camera angle.
Continuity mistake: When Sara, Mick and Zari get to the control room, Sara is wearing the sleeves of her coveralls down. In the next couple of shots she has them rolled up to the elbows, and then down again. (00:19:05)
Phone Home - S3-E4
Continuity mistake: When kid Ray opens the backpack, it's set so it's behind him. The baby Dominator comes out and takes a few steps forward, away from the backpack. In the next shot, he's barely a step away from it and it's now in front of kid Ray.
Continuity mistake: When Sara rejoins the crew at the end of the episode, Nate lifts his left hand to his face and hollers. In the next shot, he has his right hand to his face instead. (00:40:25)
Continuity mistake: After Leo makes his "heartfelt confession" to Mick and Mick says he wasn't listening, the glass skull on the desk moves position between shots without either of them touching it.
Visible crew/equipment: After Jax cuts the rope that's supposedly keeping the trap in the air, as Marty falls to the ground you can see the cable that is actually holding the whole contraption up while the rope falls a bit to the right.
Factual error: The quiz on the cereal box asks "who is the most prolific serial killer in the United States", to which Ava replies "Ted Bundy. Although in my opinion, John Wayne Gacy had way more victims than he confessed to." The box also lists Bundy and Gacy (plus the anachronism). 2 things, Gacy was in fact convicted of more murders than Bundy, making him more prolific. But, Samuel Little is the most prolific serial killer in the US and Gary Ridgway is the second. Ted Bundy would rank 6th on most list.