Factual error: The team is discussing the location of the Widmark hotel which the detective says is downtown. Beckett then looks at the map and locates it only 5 blocks from E 72nd street, which is UPTOWN in NYC. Furthermore the map shows E 72 intersecting Broadway which is on the west side of Manhattan. This entire mapping exercise is incorrect.
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.
Character mistake: Beckett identifies the area given for the Skype call as the whole city of Paris. It's hard to believe that Beckett knows the area of a foreign city by heart, the problem is also that she is wrong. Paris actually has an area of about 105 km² (41 sq mi).
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: 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)
Other mistake: Beckett points out to the murderous security guard that some graffiti magically appears on the tape, showing the live image was replaced with older pre-recorded footage. "Do you see that? It's actually kinda hard to spot." Except the graffiti doesn't appear at the point where the live feed restarts (when the guard enters the shot), it only appears on the footage after a cut away and then back.
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 .
Character mistake: When Beckett is bemoaning losing her job with the government, she says she hasn't been out of work since she was fifteen. But in "Always" at the end of season four, she is suspended by Captain Gates and chooses instead to resign from the force.
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)
Like Father, Like Daughter - S6-E7
Continuity mistake: When Castle and Alexis visit Frank at the prison in Pennsylvania, as he's brought in, the guard removes his handcuffs. In one shot, Frank's arms are flat on the table, and in the following shot, his left arm is raised. (00:08:20)
Audio problem: Ryan yells "show me those hands" but the audio happens before we actually see him move his lips, meaning it is terribly out of sync.
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)
The Good, the Bad & the Baby - S6-E10
Continuity mistake: When Lanie arrives at the precinct to check on the baby, her shirt is tucked into her jeans, as seen when she bends over to pick the baby up. When the shot changes to behind Lanie, her shirt is untucked, and then when the shot changes to the front of Lanie, her shirt is tucked in again.
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)
Smells Like Teen Spirit - S6-E15
Factual error: German bearer bonds would never have English writing on them, and furthermore instead of just saying "German bearer bonds" the company or other entity that issued them would be named.
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.
In the Belly of the Beast - S6-E17
Continuity mistake: After Beckett gets back from her assassination assignment, she washes up in a bathroom and places the gun she was given in her front waistband near her stomach. Then she goes to meet Lazarus (Vulcan Simmons) in the basement, who realizes she's Beckett. When she pulls out the gun, she reaches behind her back to get it. (00:27:25 - 00:31:00)
Plot hole: Beckett is framed for the murder of Vulcan Simmons by someone (likely connected to Senator Bracken) who has access to her backup weapon locked in a safe in her apartment. It's never really proven that she's innocent; the murder weapon is never recovered, and the real murderer is never brought to justice. And though she's still technically a murder suspect, Beckett is allowed - after finding the evidence for which she's been searching - to act in her law enforcement capacity and arrest Senator Bracken.
Suggested correction: She had evidence, (the tape) that he was a murderer and probably explained before the arrest. That should be enough to prove her innocence, and to also prove him guilty.
The tape is proof of Bracken's guilt, not of Beckett's innocence. And as we've seen, Bracken is too careful to get caught; he wouldn't clear her name just to help her, or without something in it for himself. People would start asking questions why, and the truth of his criminal wrongdoings would be exposed. And though it was later retconned in S8 that Mr. Smith helped clear her, it took two years for the writers to address it, so as a standalone episode, it's still a gaping hole in the story.
With proof on Bracken, that could also prove Beckett's innocence. It's two birds with one stone. Nobody believed her anyway and she was accused of murder, but once the news on Bracken gets out, nobody has any reason not to take her more seriously.
