I'm a huge Danny Kaye fan and have been waiting for years for "Up In Arms" to come out on DVD. (I would have been happy to see it on VHS if had been available during that era.)
All four movies are good "clean" fun. "Up In Arms", a bit of an Army spoof, was Mr. Kaye's first movie and it is a delight to see Dinah Shore performing as well. "A Song is Born", while entertaining, doesn't have any of the performances featuring Mr. Kaye's unique style and panache. It is great fun to see many of the big band heroes from the 1940s like Benny Goodman, Luis Armstrong and others while Professor Frisbee (Kaye) and the professors are writing a history of music. "Wonder Man" is a favorite. Seeing Mr. Kaye playing twin brothers with quite opposite personalities is entertaining. “The Kid From Brooklyn” is a fun boxing show with night club performances thrown in for good measure. Vera Ellen is spectacular as always.
One thing to note about all of these movies is that they are from the 1940s and this is reflected in the films. However, even if unversed on the era, you can enjoy all of these movies.