At around 16:10, when discussing about how to see the tattoo on a possible suspect, Beckett says she needs a partner to go into the women's locker room. As she is saying that, the camera is on Espo, Castle, and Ryan. All three men raise their hand to volunteer to be Beckett's partner. Ryan, while the camera is facing them, raises his left hand. When the camera switches to behind the three men and faces Beckett, you can see Ryan has his right hand up, instead of his left hand.
Towards the end, around 41:35 Beckett's phone received a text message. Castle tells Beckett that her phone buzzed. Beckett checks her phone, her shoes are in her right hand and she picks up the phone with her left hand. A few moments later, when Beckett is looking at her phone, it's in her right hand and her shoes have "magically" moved to her left when she picks up her jacket from the chair.
A paper notepad would not stop/catch a .38 Special bullet, as depicted. It would go right through.
To identify the owner of an incriminating tattoo, Beckett goes undercover at a spa with Hayley. Considering they're best friends, it seems odd that Beckett didn't first think of asking Lanie to go with her.
When Kate apologizes for being a day late for their wedding anniversary, Castle replies that they could consider it 364 days early for the next one. However since 2016 is a leap year, it would mean 365 days early in this case.