Character mistake: The day of the race, Higgins reacts to Adelaide mentioning Norman as if it were the first time he ever heard the name, but he was present when she mentioned him, and the case. Not being all that involved, technically he didn't have to remember that, but he remembered the detail of the uncle, and forgetting that the woman was talking about someone being murdered, is quite odd - and is just done for the cheap gag that follows.
Character mistake: Twirling their moustache at the success of their operation, the villains say that the heroin is worth 10 million dollars. When Rick looks at all the drugs, he says it must be worth "a couple millions."
The Curse of the King Kamehameha Club - S1-E11
Character mistake: Poolside before his mysterious illness, Rick tells Magnum that the board wants to cancel the "ski surf" race. That'd be the "surf ski" race. He should know, he organizes it. (00:33:15)
Character mistake: Mrs. Pauley says that Magnum told her that Bobby is alive "a couple hours ago", but it was the day before. (00:42:25)
China Doll - S1-E3
Character mistake: As they are driving to uncle Han, Mai Ling says that 11 other members of her family might have died for the urn. Magnum speculates "Cousin Ho." But he called him "uncle Ho" in his "should have stayed in bed" voiceover at the beginning of the episode. (00:09:55 - 00:28:25)
China Doll - S1-E3
Character mistake: Rick is supposed to be at least competent in first aid, but there's a short (and awkwardly edited in) clip with TC trying to get up and groaning in pain where the taping around his torso is so loose he shouldn't have bothered at all. (00:25:10)
The Ugliest Dog in Hawaii - S1-E8
Character mistake: Steve gets on Rick's truck and tells him "Just follow that car." However, when we see what happens through TC's chopper, the van is in the lead, and the car is the one following - probably it was footage shot with the intention to reveal Magnum's plan earlier. (00:42:00 - 00:42:40)
Thank Heaven for Little Girls and Big Ones Too - S1-E4
Character mistake: Higgins is extremely protective of any little trinket and bric-à-brac at the Estate, but in this episode he barely bats an eyelid at a post-impressionist masterpiece being handled casually, rolled up and passed around. He handles it pretty badly himself.