Visible crew/equipment: When Castle pokes his head through the doorway with his eyes closed before walking in on Alexis again, in the shot facing Alexis the boom mic hovers in front of her, at the bottom of the screen. (00:40:50)
Continuity mistake: The guy at the door says "I purchased a ton of crap at that place" and the door is partially open next to him. It cuts to a barely different angle as he says "come on in" and the door is instantly wide open. (00:25:25)
Character mistake: They find the DeLorean and the licence plate is shown: LFM-26W1. Esposito calls in and says the car needs towing: "'81-'82 DeLorean, New York plates, JLD..." Nothing like the actual licence plate.
Continuity mistake: At the end of the episode, the Mercedes S class that is burning is not the same car that Castle drove previously in the episode, it's an older model one. Either it's a mistake or has to do something with the plot resolution, decide yourselves...
After Hours - S5-E8
Plot hole: Beckett and Castle lose their cell phones and wallets and are being chased by the mob. Beckett steals a cell phone from a car but can't use it, since it is locked and then shut off remotely. Any cell phone can always dial 911, whether locked or blocked from service.
Continuity mistake: When they first arrest Penelope they show her first fingernail broken. In the next scene when she's talking about the nail, she shows her middle fingernail broken.
Audio problem: When Demming writes "Stolen books" on the white board and says, "Killed for a fortune in stolen books, so I'll run the work crews and see if anybody has priors", his lips don't match what he's saying at all.
Home Is Where the Heart Stops - S1-E7
Visible crew/equipment: When Castle and Beckett step into the elevator whilst talking about The New York Review of Books, a cameraman is reflected in the left door of the elevator as it closes and the scene ends.
Continuity mistake: When the Captain hands Beckett his flask, from behind she unscrews the cap and lifts it to her mouth. We then cut to a front-on shot of Beckett and she lifts it to her mouth a second time.
Flowers for Your Grave - S1-E1
Other mistake: Beckett and Co are in the mental patient's apartment. She opens one of Castle's novels. The title page says that it's called "A Skull at Springtime." However, on the opposite page, it says that the book is "Storm Fall." (00:19:50)
Continuity mistake: When Beckett and Castle kiss at his party at the end of the episode, the camera is behind Castle, and Beckett's head is leaning toward her left shoulder. When the shot switches to behind Beckett, her head is leaning to the right. Then it switches back to left. (00:38:05)
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.
The Fast and the Furriest - S5-E20
Continuity mistake: Beckett's sweatshirt changes position on her left shoulder throughout the scene in which she and Castle are sitting in bed discussing Bigfoot. Her sweatshirt is initially covering her shoulder and then keeps going lower and lower off her shoulder every time the camera comes back to her .
Continuity mistake: After Special Agent Shaw tells Beckett to go home when they believe a murder took place in a multi-story car park, Beckett touches her hair with her right hand. She is wearing gloves on both of her hands when she does this. In the shot immediately afterwards however, her right hand is bare and she is holding the glove in her left hand.
Plot hole: All three murder suspects claim to have seen a picture above the fridge of the hotel room. This picture however is only in the hotel room where the actual murder happened, not in the room that is shown in the video, so they couldn't have seen it.
Continuity mistake: When Kyra introduces Greg to Castle, Greg takes Kyra away to greet the new arriving guests. He places his left hand on her right shoulder twice between shots.
Under Fire - S6-E11
Other mistake: At some point during filming, Stana Katic injured her right hand. At episode's end, when Beckett hugs Esposito and Ryan after they're rescued, her index and middle fingers are taped together, and the injury doesn't appear in any other scene. (00:41:00)
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.
Nanny McDead - S1-E2
Continuity mistake: At the loft, Alexis is putting away the leftovers and clearing the kitchen. Her hair moves back and forth over her right shoulder throughout the scene. (00:10:30)
Answer: The networks are now having winter breaks. A mid season end of new episodes, when they come back after the new year, it picks up as if several months have gone by. Beckett suggested Valentine's Day as a "chick flick" to watch as she is moved in. Also some episodes are not scheduled to air in order.