To obtain reluctant witnesses' cooperation, Castle will occasionally offer to name a character in his Nikki Heat books after them, often right in front of Beckett, or Ryan and Esposito. While it might appear to be a harmless offer, any competent defense attorney could paint it as bribery and witness tampering, which are both illegal; as cops, they don't seem bothered by an action that could potentially get a solid case against a suspect thrown out at trial.
In many episodes detectives track somebody by using the entrance/exit information from their MetroCards. In NYC the card is only swept once at the entrance but not on the exit (except on Staten Island where card is swept at the exit only), so the person can travel almost the entire subway system with only the entrance recorded. Exit swipes are in use in Washington, DC and San Francisco's BART where zonal fares are in use.
Throughout the show Beckett goes on undercover operations from time to time. However, with all of the publicity associated with the Nikki Heat books that are based on her, it is more than likely her "cover" would be blown and would be useless in that role.
The goof item below may give away important plot points.
During the storyline arc of their separation in season eight, Castle fears he's lost Beckett's love and spends a great deal of time and effort trying to win her back. While not entirely unjustified, his fear is a little over-exaggerated. On numerous occasions, more than simply telling him, Beckett has demonstrated how much she truly loves him; perhaps most notably, she is willing to risk not only exposing their relationship to Captain Gates, but losing her job in the process, to comfort him when Alexis is abducted in season five.