My favorite example is Buffalo Springfield (Neil Young, Stephen Stills) appearing on “Mannix” in 1967 as the house band in a go-go bar.
mmm
Neither a borrower,
Nor a lender be.
Do not forget,
Stay out of debt.
Think twice,
And take this good advice from me;
Guard that old solvency!
There’s just one other thing,
You ought to do;
To thine own self be true.
I loved the scene in Umbrella Academy where (for reasons that are spoilerish) the Cast appears to break into a song battle:
And oldy by a goldie though, from the TV Series Lexx, is the Brigadoon episode:
Sometimes you ask just the right question, and sometimes magic happens. 
Thank you, and keep them coming.
“Third Rock from the Sun” did an episode in which the alien characters experience dreams for the first time. Harry (French Steward) does this:
I remember Madness singing “Our House,” my friend thought it was the funniest thing when the saxophone gets smashed into the camera.
Hal’s barbershop quartet, the Gentleman Callers, got some screen time on Malcolm in the Middle:
Bruce Willis was often willing to sing on Moonlighting, even when he shouldn’t have, but I’ve always liked this one from “Atomic Shakespeare”. In fact, I like it so much that I played it at my own wedding ceremony, although there was less bondage involved:
The underrated (in my opinion) Psych featured a musical episode.
ETA: The Simpsons, Family Guy, Futurama, and American Dad all have moments where they break into song.
It’s Always Sunny in Philadelphia had “The Nightman Cometh”:
Slade were a huge band in the UK. Not so well known in the USA other than as “that band that did the video with the gurning” or Quiet Riot did a couple of covers of their songs.
Lead singer Noddy Holder retired from the band and did some acting including the nostalgic TV comedy set in the 1970s “The Grimleys” where he portrayed a music teacher called Neville Holder. Neville being Noddy’s real name and the character was basically an alternative reality Noddy as if he had never been a Rock Star.
In an episode where Neville regrets his life choices he does this epic unplugged version of one of Slade’s best known songs.
TCMF-2L
Perhaps my favorite thing ever
It really is a wonderful thread. 
The Dick Van Dyke Show had many musical interludes. The one that comes to mind is The Twizzle S1, E23, with Jerry Lanning singing the title song and “This Nearly Was Mine”. Plus, it had Classy Freddie Blassie at the very end.
Come on get happy?
I was wondering why no one had mentioned House yet, what with the scene of House leaving the wedding and going to rehab to As Tears Go By, not to forget Maggot Brain.
And at the end of the episode Skipper flips Gilligan out of his hammock (because he won’t stop singing it) and says “and this is the answer that you get from me!’
My brain loves you, @SpinyNorma.
I was gonna mention the music being Carmen. But I didn’t wanna seem too esoteric.
I have my rep as a dumbass to maintain. Ya know.
I mean we could get really silly and say musical interludes in shows centered around fictional families or goof balls(Monkees) being pop/rock bands have these weirdly placed musical numbers. But the title says non-musical shows. So that’s what I was going with.
I am so glad to hear from others (Sorry, I’m too lazy to scroll back up to get the names) who remember the Gilligan’s Island musical version of Hamlet! I honestly have never come across anyone who did and have gotten some weird looks when the subject came up, the sort of looks that suggest I need either more medication or less.
I had actually started to think I was perhaps having a weird hallucination from childhood when I accidentally saw a rerun of the episode. There were, of course, no witnesses.
Thank you, Beckdawreck! I don’t post often but my brain enjoys reading your assorted tales.
I don’t recall when I heard Carmen can made the association but I remember my reaction as I realized why the music sounded so familiar. It was definitely a head-jerk, jaw-drop moment.
You’d fit right in at my house. My older Brothers and Sister well into to their 60s do a stirring rendition of the Gilligan ‘Hamlet’ every reunion or gathering.
It does help they all have liquid courage at these occasions.
It’s roundly appreciated by the youth around here and Me.
(And TY for the compliment)