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)
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: Alexis has just walked into Castle's study and is relaying a message from his publisher to her dad. She holds the note differently in consecutive shots. (00:30:20)
Continuity mistake: Castle asks his daughter what she would do if she needed a quarter of a million dollar, just like the murdered victim. She answers that she would ask him for it. Watch her necklace. It jumps in and out of her top. (00:31:10)
Continuity mistake: When Beckett is interrogating the drug dealer, in shots from behind, only her right hand is on the table. In shots from the front, both of her hands are on the table. (00:09:20)
Continuity mistake: During breakfast at Castle's loft a jug full of orange juice disappears off the kitchen island when Castle answers the phone. (00:01:40)
Continuity mistake: When Kate gets a call from Lanie who asks her to come down to the morgue, Kate's jacket is beige. When Kate enters the morgue, it's black/dark blue. (00:07:40)
I, Witness - S7-E13
Continuity mistake: Beckett has just walked into the P.I. office with two coffees to meet Castle. The way she holds the coffee in her right hand changes instantly. At first, she's holding the top of the cup, then she's holding it on the bottom. (00:31:15)
Trivia: When Ryan and Esposito are searching for Jones the bodyguard, the man they first approach is Kris Brkljac, to whom Stana Katic has been married since 2015. (00:05:35)
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: At the end, Beckett spills coffee on the letter. In the first shot, before she frantically blots the spilled coffee, it says that the writer is "a junior at P.S. 370." Afterwards, this has changed to "P.S. 311." (00:40:40)
Continuity mistake: When Castle and the flight attendants are attempting to obtain fingerprints from a suspect, the liquid in the cup that he throws away changes colors. (00:16:25)
Child's Play - S7-E4
Continuity mistake: When Castle is allowed to talk to Mrs. Ruiz's second graders, as he enters the classroom, he is hit in the temple with an object thrown at him and holds his hand to his head in pain. After he turns to say something to Beckett, as he turns back, not only has the hand on his head switched, it's also much lower, down on his chin. (00:07:55)
Home Is Where the Heart Stops - S1-E7
Continuity mistake: When Esposito pulls up Evan Mitchell's rap sheet, it says he is 5'9", but in Mitchell's mugshot, the top of his head crosses the 6' mark.
A Chill Goes Through Her Veins - S1-E5
Continuity mistake: When Castle and Beckett visit Mr. Davidson, he comes out chasing his two grandchildren - a tall girl in a red jacket and another one who's slightly shorter in a light pink jacket. When the detective and Castle walk up, the grandkids have switched places. When Mr. Davidson says, "Did I scare you," Beckett has her hand in her jacket, but when Beckett says, "Mr. Davidson," her hand is not in her pocket anymore. When Mr. Davidson says, "Why don't we go inside," Beckett is putting her badge in her pocket, but in the shot before she already put her badge in her jacket. Finally, when Mr. Davidson says, "Alright!" the girls have switched places again between shots. (00:09:50)
A Death in the Family - S1-E10
Continuity mistake: Episode 1-10, "A Death in the Family": when Agent Sorensen offers Beckett a part of the sprinkled donut, the location of where he holds it changes between shots. (00:23:50)
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.