After Beckett struggles with the bad guy, he throws her over the side of the building. She manages to catch herself on a ledge by her fingers. When the camera is shooting her from the side, the buildings fire escape is to her right. When the camera is shooting her from above, the buildings fire escape is to her left.
As the camera is panning the framed pictures in Captain Montgomery's house, it shows a family picture (he, his wife, his son, and his daughter). It is revealed in the season 3 finale (Knockout) that Roy has 2 daughters, not a daughter and a son. In the season 3 finale, it shows him kissing his 2 daughters and his wife goodbye as they leave for school.
In the argument between Castle and Beckett he says that Beckett has known he loves her for about a year. However, two shows earlier she tells her shrink that Castle told her he loved her seven months ago. Either the first 21 covered those seven months and the last two covered five months, or there's a "glitch" in the story line.
When we see Beckett hanging from the roof the fire escape metal stairs are to her right. when she looks down they are to her left and to her right we clearly see the building's corner with some grass on the ground. she still is hanging her body facing the building so it should not change.
At the initial crime scene, the M.E. says the victim was shot in the shoulder with a .357 and in the head with a 9mm. In reality, these two bullets are .002 inch different in diameter, so there is no way anyone could look at a wound and tell the difference.
When Castle gets a call from the 12th Precinct his phone has no service, yet he is still getting the call.
The climactic scene - supposedly set in NYC - clearly takes place atop the Rosslyn Hotel, a Los Angeles landmark.