When Esposito calls Beckett and Castle in to watch the CCTV from the gym, Beckett walks over and turns before sitting down (wide angle view). When the closer angle turns to her, she turns and sits again.
When Beckett and Castle first talk to Manny (at the Games when he's in his booth signing autographs), Beckett's pony tail changes positions with the camera shot. When the camera is behind her, the pony tail hangs straight down, but when the camera is filming her face, the pony tail is over her shoulder and in front. Due to the style of pony tail (one single curl), her hair couldn't be in both the front and the back at the same time.
At the original crime scene, Kate whips her head around multiple times, which makes her hair come out of it's single curl style. Later in the scene it's back to being perfectly styled without her fixing it.
When the motorcyclist brakes hard in the very beginning, you can clearly see the skid marks identical to his with rear wheel drifting to his right. This clearly was not the 1st take they did of the stop.
In the first game of Scrabble, Castle an avid Scrabble player is left with the letters H(4) O(1) I(1) V(4) L(1). There is an open Y(4) left on the board, making it possible to write HOLY(10) with a double word score tile. Even after subtracting for his remaining letters that would net him 15 points, more than the 9 he needed.
Castle is an avid Scrabble player who has 'never' been beaten. So, when Beckett plays QUIXOTIC and asks "How many points is that?" Castle would be more likely to answer correctly - 356 for the word. [Q=10, X=16 (double letter), C=3, and U,I,O,T,I=1 each for a total of 34, times 9 (triple word score twice) = 306, plus 50 bonus points for using all 7 letters = 356 points for the word QUIXOTIC.] Note: Since her word score alone was higher than Castle's entire game score of 312, it makes sense that he'd be too stunned to say the final score (KB's 175+356=531 for the game).