The Case of the Red-Faced Thespian - S4-E12
Plot hole: Magnum is incredibly slow to unveil the charade at the base of the episode; he does it based on something marginal (Robin Masters does not do "research" for his novels), but he completely overlooks the key plot point of 1-17 "J. Digger Doyle" was that he does not use a typewriter, but he dictates the novels.
The Case of the Red-Faced Thespian - S4-E12
Continuity mistake: Magnum just met the Athertons. The secretary asks him to speak in private, and Mrs. Atherton gives him a knowing look. She holds her foulard in hand differently between the two shots. (00:09:10)
The Case of the Red-Faced Thespian - S4-E12
Continuity mistake: When Magnum asks Hayes what's the secretary doing at the estate and not in Greece, he has both hands on the card. Next shot and, like in previous ones by the same angle, his right hand is on the piano. (00:07:55)
The Case of the Red-Faced Thespian - S4-E12
Continuity mistake: Archer Hayes shows Magnum the invite with the wrong time. He hands it with his right, which turns into the left in the reverse shot. (00:07:40)
The Case of the Red-Faced Thespian - S4-E12
Revealing mistake: Magnum arrives at the Nest. He opens the door and moves past two swashbucklers to talk with the person playing the piano. Is he really playing the piano, though? Apparently not, because the instrument is open and you can see that the inside does not move at all. (00:07:00)
Other mistake: Throughout the entire run of the series, the very first shot of the opening contains a mistake; the helicopter's tail rotor is on the right side (mirrored), going then to its actual position (left side) for the rest of the sequence.
Revealing mistake: In the night shootout, Magnum cocks his pistol in a close-up that is obvious performed by a trained stunt double. Whoever is doing this is wearing the same clothes Magnum has but with the T-shirt in pristine condition straight from the washing machine, while Tom Selleck just took a dive in the dirt. Moreover, he wears the team ring upside-down. (00:22:50)
Revealing mistake: In the opening of the second third of the season (the one with the iconic Mike Post theme) there is a brief scene with a car propelled in the air by an explosion. It's an outtake from the pilot movie, since the banged up car is very recognizable, but it's also easy to see why it was an outtake; you can easily see cables used to make the car fly in the air. Also, that car was not in an explosion in that episode.
Revealing mistake: In the final car chase, the military police radios for reinforcement; in the exterior shot that follows the one inside the limo with Rick on his back, foot on the cockpit, you can see quite distinctly the two stuntmen meant to be Magnum and Rick, and totally not looking like them. (00:46:30)
Continuity mistake: The plane at the beginning of the episode stops with the propeller at 7:05. When Louise DeBolt walks past the propeller to get in the cockpit, the propeller is at 11:25, and goes back to 7:05 next time it's shown. (00:02:10 - 00:03:10)
Continuity mistake: During the paperboy run, TC stops the van and asks Benny a question. Benny was holding the paper vertically, but his arm is down in the next shot, and up again next, and so on. (00:32:40)
Character mistake: The coach of the Jororo team makes a joke about being cut by the king "Like Robespierre did to Louis XV in '89." That would have been Louis XVI (and not in "89" but (17) 93, although the year of the French Revolution is correct). (00:25:10)
Revealing mistake: The prince and TC's protege have a fight when the spoiled kid is eliminated. They are supposed to be angry and serious, and the teammates too have worried faces, but in the shot when TC tells him that he's out of the game, against any logic the young actors are caught laughing. (00:23:20)
Continuity mistake: Magnum gets back to his pad and of course the kid everyone's looking for just crashed there. In the first shot, overhead, you can see that Magnum's coffee table features several magazines, and in the corner, at the same level, a bowl. Magnum fetches a can of beer and sits by Danny, but the magazines now are in a different line, with the bowl above them. (00:13:20 - 00:13:50)
Squeeze Play - S4-E7
Continuity mistake: During the poker game at the beginning, the positions of Robin and Buzz's cards change a few times - Buzz unveils his winning 4 Ks to the left of the deck, but it is to its right immediately after, and Robin's cards are on or a bit away from the table's border. (00:02:30)
Other mistake: Magnum and Vivian are not listening to the TV at loud volume (they have a quiet time watching and talking without raising their voice), but Higgins, all of a sudden hears it all the way from the main house and is greatly bothered by it.
Thank Heaven for Little Girls and Big Ones Too - S1-E4
Plot hole: "Three Tahitians", one of the masterpieces from one of the most famous post-impressionist painters in the world, is authenticated by an ordinary school teacher. Because that's the person for this multimillionaire job, obviously. (00:25:20)
Continuity mistake: Leaving the hospital magnum has a black eye on his right side. When he gets home it's on his left eye.
Transitions - S8-E11
Visible crew/equipment: While Luther is walking with Suzi and Cleo sharing his "weapon, opportunity, motive" ideas, just as he says, "That is rule number two," the boom mic with its fuzzy windscreen pops up into view in front of Suzi's knee, at the bottom of the screen. (00:25:45)
Continuity mistake: When Magnum is with Dave at the radio station requesting his help as an audio engineer, Magnum's wearing the white Polo Ralph Lauren safari animal print shirt, and when it cuts to Magnum at the security office at Lee's building, now the animal print pattern is different, though he's still wearing the same style shirt. (00:28:40 - 00:29:55)