Between the 1921 Washington Naval Conference (which effectively limited the international arms race for over a decade) and the start of the crank up before World War II, the U.S. Navy co-operated with Hollywood studios on a regular basis providing locations and facilities for dozens of major and minor films showing off the country's ships (both sea and air) and service men to keep them in the minds of the general population as the admirals fought for ever constricting budgets with an isolationist minded Congress. Also well worth checking out are THE FLYING FLEET and HERE COMES THE NAVY (in which feuding Jimmy Cagney and Pat O'Brien serve everywhere from the Arizona to the airship Macon - both to meet famous ends in later years).
MURDER IN THE FLEET may be among the least of these info-tainment efforts, but from the golden age of the classic murder mystery, it offers an enjoyable, more or less workable plot, an "about to be 'A List'" cast and some fascinating shots of actual elements of the U.S. fleet (the entire film is set on the USS Carolina aside from a few studio interiors and includes a number of exteriors of the ship under way including the actual crew).
Of special interest may be Key Luke's fleeting appearance as an aide to an ambassador/suspect from a (renamed to avoid "offence," but made-up to leave no doubt) fictionalized Japanese Empire. Someone might do a fun afternoon's mini-film festival of "Key Luke Afloat" with this, the 1936 ANYTHING GOES (Luke played one of a pair of gambling "Chinamen" on a civilian Atlantic crossing with Ethel Merman and Bing Crosby - a role somewhat reduced from the Broadway original) and (a year later) in his most famous role as Charlie Chan's "Number One Son" in CHARLIE CHAN AT THE OLYMPICS where Luke travels to Europe (these were the Berlin Olympics where Jesse Owens won Gold to Hitler's great displeasure and film clips of that race are included in the movie) by ship with the U.S. Olympic Team (he's competing as a relay swimmer)while his "Pop" rushes over on The Hindenberg.
The biggest "special effect" in MURDER IN THE FLEET is probably the flooding of a powder magazine with the film's hero in it - tame stuff by modern standards, but pretty exciting as played nonetheless. The film's McGuffin, the scientific equipment being installed on the Carolina, is pretty much science fiction (though finally, 70 years later, becoming less so), and as shown in action in the film it looks a bit silly, but it was good enough for its day, and in the spirit of the style of the film it doesn't distract.
Minor, but fun. Worth a look.