Others can bring up special musical episodes of usually non-musical shows, but my example goes back a bit further: Billy Mumy and Marta Kristen singing “Sloop John B” on an episode of “Lost In Space”: Billy Mumy & Marta Kristen Perform “Sloop John B” on Lost in Space
What have y’all got for me?
Resident Alien had a lovely karaoke duet of the sheriff and the deputy singing “Wind Beneath My Wings” (a song I usually hate). Corey Reynolds, who played the sheriff, starred in Hairspray on Broadway.
Star Trek: Strange New Worlds S2E9 Subspace Rhapsody is an excellent example.
The cast of 30 Rock performs “Midnight Train to Georgia”
(and then gets yelled at by Gladys Knight) 
1965: The Standells playing a Beatles song(?) on The Munsters
I always liked the Hamlet musical on Gilligans Island.
Also Lovey, Ginger and Maryanne as the Honey Bees girl group.
The best example ever was when Ponch sang Kool and the Gang’s “Celebration” on an episode of CHiPs:
Dominic Chianese (as Uncle Junior) performing an impromptu rendition of “Core 'Ngrato” at the end of the Season 3 finale of The Sopranos:
Lt Uhura breaking out in song in several ToS episodes and once Spock on the notorious (2nd worse) Space hippies episode
Louise (Yeardley Smith) sings “Cry Me a River” on Herman’s Head:
Eugene Fodor plays Bazzini’s “Dance of the Goblins” on SCTV (4:02):
The Goblin Song on Doctor Who.
Spock didn’t sing in “The Way to Eden,” he just jammed on his Vulcan lyre. He sang “Bitter Dregs” in “Plato’s Stepchildren,” one of my candidates for the absolute worst ST episode of all time.
The Dickies on CPO Sharkey - one of the first appearances of punk rock on network TV:
The Darlings on Andy Griffith or Opie singing. So cute.
(I agree, it could be said The Andy Griffith Show might’ve been a semi- musical show)
Spock jamming in “The Way to Eden”:
Somebody’s going to bring out “Once More With Feeling” from Buffy the Vampire Slayer, but I think Anthony Stewart Head had some brief musical interludes in non-musical episodes.
And the only scene I remember from the unmemorable Who’s the Boss was a date with Angela’s high school crush, played by Lee Ving of Fear, who spontaneously jumps up and belts out a down and dirty blues number during their date.
The Sacred Cows on Get Smart
Pushing Daisies was non-musical, but when you have Kristen Chenowith and Ellen Green in the cast, you have to use them.
With Tucker McElroy, lead singer and driver of the Winnebago.