As the new kid on this block, and a Judy Holliday fan too... just a comment that she did have a singing career. :) She won the 1957 Tony for best actress in a musical for Bells Are Ringing, and starred in the film version (with Dean Martin). :)
Her showbiz career began as a member of the nightclub group The Revuers. The group included two future titans of musical theater/movie musical lyric- and book writing, Betty Comden and Adolph Green, who were also wonderful performers of their own (and others') material.
Does anyone know where this clip originated? It's utterly delightful.