Flowers for Your Grave - S1-E1
Other mistake: Beckett and Co are in the mental patient's apartment. She opens one of Castle's novels. The title page says that it's called "A Skull at Springtime." However, on the opposite page, it says that the book is "Storm Fall." (00:19:50)
Other mistake: It is said G. Callen has lived in 37 foster homes between the ages 5 and 18, the longest ever having stayed in one is 3 months. 5-18 is 13 years, meaning even if he had lived 3 months in EVERY foster home, it'd have to be a minimum of 52 foster homes. Since 3 months was the longest, it is logical that the others were shorter therefore the number would have had to be much higher.
Other mistake: When Neil is in prison and has time on his hands, he puts pieces together that were shredded. Neil's name in the file is CAFFERTY, not CAFFERY. Also, in the same scene, the words "e've" can be seen with the W missing. Yet, later on, when Neil shows Burke the final photocopied piece, the W that wasn't there is now visible.
Other mistake: Regarding the suicide of Karen Anderson, her boyfriend explains that Karen hadn't been studying with him because she wanted to be in the sun all day, but all the footage of the bridge she jumped from shows it to have been a mostly overcast, gray day. The first shot of the bridge is at the very beginning of the episode and is repeated several times throughout.