Abbott and Costello
Starring Bud Abbott and Lou Costello, the zany comedy of The Abbott and Costello Show graced the airwaves from 1942 until 1948, after a short stint as summer replacement for The Fred Allen Show in 1941.
There were often tongue-twisting or otherwise confusing dialogs ("Who's on First" is the most famous) and Costello was always getting into trouble. The vaudeville-style repartee kept the live audience in stitches.
In the tradition of the day, there was always a song, courtesy of Connie Haines or (later) Marilyn Maxwell.
You can find more episodes in the Ear Benders section.
Here is a sample of the online episodes currently available at The Old Time Radio Fan:
Stolen Oranges, Lou Shoots Orange Picker
1945-02-01 (9.91 MB) [28:51]
Lou Hides From His Girlfriend
1945-11-15 (9.9 MB) [28:51]
Lou the Fireman
1945-12-06 (10.07 MB) [29:20]
Babysitting Job
1948-04-28 (13.52 MB) [29:32]