Revealing mistake: When Castle arrives at the murder scene and views the victim's body, the "corpse's" eyelids are twitching.
To Love and Die in L.A. - S3-E22
Visible crew/equipment: At the pool, after Beckett makes a stunning entrance from the water, a white reflector screen is seen on Ganz's sunglasses. (00:31:40)
Plot hole: Brandt sabotages the mercenaries' explosives so they are killed when trying to escape. No real team of professionals would let the rich yahoo who hired them touch their gear. Nor would they use any explosives he provided without (at the bare minimum) inspecting them first. It's a plot hammer to wrap up what happened to the mercs so they can move on to Brandt's capture.
Suggested correction: So, they knew they would be framed/killed at the end? Also, if he was paying or threatening them, then why wouldn't they listen to him?
Walk up to a professional you have hired and is doing his job. Plumber, AC tech, whatever. Ask him if you can mess around with his stuff while he's working. What do you think he'll say? Especially if your messing with it could cause harm to him.
Continuity mistake: The paper dolls found at the crime scenes are mixed up. The 1st one found was from Wyatt's scene and called the "Persecution of Kings" which would lead them to King Street, the site of Wyatt's murder. This one should have been found by Sara Vazquez.
Continuity mistake: Jennifer Beals' blouse mysteriously changes from white to blue while they are in the CIA secret bunker prior to heading out for the final confrontation.
Continuity mistake: When Castle comes home to Sophia waiting for him in his office, she is holding the Derrick Storm graphic novel; it keeps changing from open, to closed, even to her closing it every time we see it in her hands. It's very noticeable.
Headhunters - S4-E21
Continuity mistake: During Shea's interrogation scene, he has blood running from his head down to his eyebrow which collects into a blob. Between shots, this blob of blood disappears and reappears.
Cloudy with a Chance of Murder - S5-E2
Continuity mistake: When Beckett is trying to get dressed for work at the beginning, she takes off a blouse she deems unsuitable and the bullet scar on her chest, which is usually at the start of her cleavage line, is missing. (00:00:25)
Cloudy with a Chance of Murder - S5-E2
Continuity mistake: When Beckett, Ryan, and Esposito are discussing Castle's date with Kristina Cottera, Beckett's fork is full of food in a long shot, which disappears in the following close-up without her eating anything. (00:32:15)
Continuity mistake: Castle and Beckett are talking about the case in the precinct and his coffee runs out. Beckett offers to go refill the mug. She grabs his white mug in her right hand. In the next shot, it is now in her left hand. A blue mug is in her right hand instead. (00:05:15)
Factual error: On the way to the Hamptons, Castle's car is driving on the left side of the road, even though they're in the US (where people drive on the right side of the road). (00:03:10)
Revealing mistake: After Castle and Beckett are arrested for interfering in Chief Brady's investigation, a deputy tells them that Brady wants to speak to them; Beckett stands up to leave the holding cell, but is held back by the pair of handcuffs chaining her to the bench. The cuff is not actually on Stana Katic's wrist; she's just holding it in her hand.
Continuity mistake: Castle gives Beckett a coffee mug in a very loving way by moving her hand to hold the mug by the handle while she is reading a file and not looking. Then in the very next scene she is holding the mug on the bottom in her palm. (00:05:24)
Character mistake: The killer's motive for the murder is getting kicked out of the band akin to Beatles drummer Pete Best, and Castle whispers to the camera that bass player Stu Sutcliffe would have been a more appropriate comparison. Sutcliffe quietly quit the Beatles voluntarily, while Best was fired, with his dismissal becoming widely-known public knowledge, so the comparison to Best actually is more applicable.
Secret Santa - S5-E9
Other mistake: When Ryan comes into the precinct to tell everyone that the victim did not have job booked on the night of his murder, that fact is already written down (in red) on the timeline board in the background,. (00:11:00)