In the episode about the gangster and his moll, Clark Kent, not Superman, is shown using his super powers in his civilian guise - breaking down doors, roughing up bad guys, and grinning as bullets bounce off him as Clark.
"Five Minutes To Doom": Usually bumbling comic-relief Jimmy Olsen's vacuum cleaner salesman stunt works perfectly. He gets into the Villain of the Week's office, gets the information he came for, and gets out completely undetected.
"Jimmy Olsen, Boy Editor": Lois sweet talks one of the mooks into letting her see how heavy his machine gun is and swipes it. When the leader claims she doesn't have the guts to shoot somebody even if she knew how to use it properly, she admits he's right. So she empties the gun into the ceiling, then calmly tosses the now useless weapon back to its owner ("It's even lighter with all the bullets gone."). The looks on the villains' faces perfectly capture how wild it is watching Lois Lane of all people firing a machine gun. She and Jimmy would have been able to walk away unscathed with no Superman rescue necessary if not for a Diabolus ex Machina.
The radio series had a meta crowning moment with the Clan of the Fiery Cross arc, where Superman took on the KKK: armed with real KKK secrets obtained from Stetson Kennedy, it did so much to strip the mystique and perceived "respectability" from the Klan that it managed to seriously reduce the organization's recruitment and derail its then-current rise in popularity.