In one episode of Adventures in Odyssey, Ed tries to cut Marvin's hair and botches the job. He is reminded at least twice (before and after the botch job) "Remember the snowflakes?" as a reason as to why he shouldn't be let near a pair of scissors. We never find out what exactly happened with snowflakes.
In the earlier episode "A License to Drive", we find out bits and pieces of what happened during Eugene's practical driving test but not all of it and all we know as the overview is that it was so bad that the instructor forcibly took control of the vehicle, drove back to the DMV and resigned.
In one episode of The Goon Show, Henry Crun makes a passing reference to "that terrible night I drank the dandelion wine."
Done by an Omaha station talk show called Todd & Tyler, which is of course hosted by two guys named Todd and Tyler (they're mostly considered shock jocks, in that they do their best to offend as many people as possible). A couple years ago, they had a Running Gag where, whenever someone said the phrase "two weeks", they'd start laughing like crazy. They refused to explain why a period of time could incite such hilarity on air, though supposedly, if one met them in real life and asked for the meaning behind it, they would be kind enough to explain. Though nowadays they don't seem to find the phrase nearly as amusing.
In the 80's WUSB Stonybrook comedy radio theatre The Tripod Family, there was an "incident" in Big Bear City which forced the family to flee to Long Island. Any time they are close to revealing what happened something happens, the show cuts to another scene or they use the "You don't want to know" line, but it is inferred that one of the Uncle's business plans was to blame.