"Big Band," by Loren D. Estleman, in Alfred Hitchcock's Mystery Magazine, January/February 2012.
I have admitted before that I am a sucker for Estleman's stories of the Four Horsemen, the racket squad of the Detroit Police Department. These not-very-heroic heroes are unloved by their bosses but are determined to keep their jobs, and thereby stay out of the armed services. The historical detail is perfect and the language is witty and snappy.
This story centers on the leader of the group, Lieutenant Zagreb, who is not in the war because of a heart murmur: "it kept murmuring Don't go." He gets a special request from an ex-sweetheart: look after her trumpet-playing lover while she goes off to serve in the WACs. Turns out the lover is a bad musician and an angry drunk. Pretty soon there's a murder to solve.
Did I mention the witty language? Here is a random line, describing a cop named Canal: "He smelled one of his thick black cigars -- no one ever said he wasn't a brave man -- and put a match to it, clouding the air with the stench of boiling bedpans."