Six Paintings, One Frame - S1-E4
Character mistake: The art expert says that lapis is a rare gemstone and was used in painting only up to the Victorian Era, after which became too expensive. That's quite misleading; it's not that lapis became more rare or expensive, it's just that adequate artificial substitutes became available.
Death Is Only Temporary - S1-E6
Character mistake: The web page in memoriam of Elizabeth Cole is "In Memorium." (00:13:10)
Character mistake: Magnum and Higgins never retrieve the large and conspicuous device they used to hack the 'top level firewall' of the hotel database - and they even put it in a frontal USB port.
Character mistake: When they are at the docks/harbor and they capture the person who poisoned the guy that hired Magnum, Higgins tells him if he resists, Magnum will shoot him. She says he has 16 bullets, but he's carrying a 1911 and the most it will have is 10 rounds.
Dark Skies - S5-E8
Character mistake: Higgins and TC are investigating a drug smuggler camp in Maui. TC sees a box of cigarettes and says the people in the camp must be Latino because the cigarette box is in Spanish. Wrong. The box says Cigarillos, which are small, short cigars. If the box was in Spanish it would say Cigarrillos, which is cigarettes in Spanish.