Who Done It finds our intrepid duo as soda jerks who want to be mystery writers and work on the Murder at Midnight Show. Bud and Lou go to a broadcast just in time to witness the real murder of the head of the network, Thomas Gomez.
So what do our two geniuses think to do? They decide to impersonate police officers and try and capture the criminal themselves. Interfering with a police investigation is an offense unto itself, but when Abbott and Costello do it, it's strictly for laughs.
The two cops who don't think it's that funny are William Gargan and William Bendix. Supposedly Costello was not happy with Bendix playing the dim bulb detective because he was getting more laughs than him. Mary Wickes who plays the secretary of Thomas Gomez also said she did not get along with Costello on the set.
The actual murderer turns out to be a very peripheral character who only had a couple of inconsequential lines before he's unmasked at the end. I tend to think there was probably more of his part, but it was edited out. Not that there are not a host of suspects like Patric Knowles, Jerome Cowan, Don Porter, Ludwig Stossel, and even Wickes and Louise Allbritton.
One actor who did not get any billing, but should have because he was very funny constantly getting the better of poor Costello was Walter Tetley who played the fresh mouthed young elevator operator. What he did to Costello bordered on sadism.
Who Done It is a fine slapstick burlesque of all these mystery films that all the studios were putting out back then. I guess it said that if even Abbott and Costello can solve a case anyone can.