Continuity mistake: When Danny dives into the ocean he's wearing different shorts than he was on the beach.
Plot hole: There were 3 gunshots. 1 from Magnum, 1 from the killer and 1 from the victim. Yet they arrest Magnum with checking the bullet striations. He would be declared innocent almost immediately. (00:17:00)
Plot hole: Barney Kale decides to reopen a case he couldn't solve. Based on the outcome it makes no logic whatsoever. He is the murderer. Why would he bring attention to it put himself in danger? (00:46:50)
Suggested correction: He was being blackmailed by Gary Roberts. He "reopened" the case as a ruse to kill Roberts.
Death Takes a Curtain Call - S1-E9
Revealing mistake: When the cops arrive at supper time at Jessica's door, the background is a blatant painted backdrop. (00:31:45)