I went home at 3am, and put on Castle. Watched around 10 episodes, to see the patterns and try to solve the murders based on script logic.
Castle uses a basic three act structure. There's the cold start opening, first act, where we establish Plot A and Plot B. As well as the murder scene, and introductions. Almost all characters in the episode will be introduced in the first act.
You can glean enough information and suspects in the first act to make a 90% accurate guess as to who the killer is.
1. A major witness or character given a spoken role, but was only featured briefly. Speaking roles get paid better than extras. They wouldn't just hire an actor and pay him to speak three lines. If anyone spoke between three to seven sentences, and they just wander away from him, THAT'S the murderer.
2. First suspects brought in for interrogation are almost never the killers. If they were, then the episode is a special or will last just 15 minutes. There needs to be at least three suspects questioned, either in custody or at large, as these people need to unearth dirty little secrets they have been hiding.
3. Some clues will be withheld until the last act, during the reveal. So don't count on the evidence. Just the suspects.
You can apply this kind of thinking to any episodic crime show and discover the murderer. Except for Law & Order, which usually follows a more linear approach to investigating, only unearthing the prime suspect at the conclusion of the investigation.
The other shows like The Mentalist, Castle, CSI: Miami, whatever, uses the normal approach.