As Jimmy sets up his fruit orchestra, the third pineapple (string bass) he places falls over. As he finishes conducting and the camera pulls back, we see the fallen pineapple standing up perfectly.
Jimmy Conners claims to have gone to Chicago for the World Series three years before. The World Series is always played at the home stadiums of the competing teams and the 1937 series was between the New York Yankees and the New York Giants. The 1938 World Series was indeed between the Chicago Cubs and the New York Yankees, but that would have been only two years before the film was released.
When the June Preiser character's dad hires a train to get Mickey, Judy and their band to Chicago for the Paul Whiteman broadcast, it's supposedly a surprise. Yet in the train station send-off scene that follows, presumably later that same day, the caboose is festooned with a "Chicago Here We Come" banner and several people in the crowd are carrying professionally made picket signs. In 1940, it would be all but impossible to produce those signs on such short notice.
In an anachronism typical of movie musicals of the time, when Mickey and Judy's band wins the competition, it is announced that musicians, singers and dancers from the various competing orchestras will perform an impromptu big number together. Even though they've never met before, nor rehearsed, nor had even five minutes to create sets and costumes, everything comes off seamlessly and with M-G-M's usual level of polish.