Continuity mistake: When Beckett watches the weird/creepy video and a few images after seeing the ghost at the house, she hears a little girl laugh. However when Castle watched the video, that laugh happened at the moment with the ghost at the house, not after.
Continuity mistake: Alexis' necklace changes in almost every single shot that it's shown. Sometimes it's a single loop; sometimes it's two; sometimes it's three; sometimes the shade of red is darker or lighter; sometimes the loops are big or small and so on. It's very noticeable.
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.
Continuity mistake: In the opening scene, when the rider is knocked off his bike by the car, the bike ends up under the front wheel of the car. When the perpetrator drives off, the bike is some distance away from the car.
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: 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: 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)
Nanny McDead - S1-E2
Continuity mistake: At the precinct as they are watching the security footage from the elevator, a small bit of Beckett's hair keeps moving back and forth over her right ear throughout the scene. (00:06:55)
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)
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)
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)
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)
Backstabber - S8-E18
Continuity mistake: When Wesley changes his statement after Hayley gives him a "swirly", his wet hair changes position between shots.
Death Wish - S8-E17
Continuity mistake: Beckett keeps a jar of nuts on her desk. When she's talking to Castle about Lars, she reaches over her desk to put the jar down. But in the following shot when Castle stands up, both her hands are suddenly in front of her laptop, clasped together.
Death Wish - S8-E17
Continuity mistake: When Castle beings theorising that Genevieve might be a genie and that she wants him to be her new master, Beckett is holding a glass in both hands. In the following shot, she's holding it with just one hand.
Continuity mistake: When Castle gives Beckett a mug of coffee at the beginning of the episode, she isn't holding it by the handle, but in the shot immediately afterwards she is.
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)
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.