I’m inspired to start this thread because I was sitting here with the kids this morning watching Sesame Street, and who should appear but James Blunt, singing “A Triangle” to Telly Monster.
I also looked up my other favorite musical number from Sesame Street, namely REM doing “Furry Happy Monsters.”
Just felt the need to share.
“Duck, Rubber Duck duck duck, rubber duck…”
Anything with the Martians.
“Yup yup yup. Cat. Caaat. yup yup Miaow.”
The Martians, Mah Nah Mah Nah, the Hungry Cat. Also, Celia Cruz’ songs she sang, didn’t she do numbers and the alphabet?
For the Muppet Show, Alice Cooper’s visit as well as Vincent Price’s visit were both neat. As was their rendition of Kenny Rogers’ The Gambler. Also, the Viking Longboat skit In The Navy.
The “1 2 3 4 5 6 7 8 9 10 11 12” pinball cartoon.
The Martians. “Book book book book…”
The clumsy baker. “6 Birthday Cakes! Oof.”
Kermit the Frog’s fairy tale News Flashes.
Kermit’s News Flashes, and Grover the Waiter.
I always felt sorry for that guy.
Oscar the Grouch being visited by his neat-freak grouch friend, Felix.
For whatever reason, possibly the pensive minor key, I have always been entranced by “We All Live in a Capital ‘I.’” [“Rubbing it here, scrubbing it there.”] Could just be the narcissism I like.
I also enjoy “Wanda the Witch.” [“I am a weasel!”]
Ha! Is this real or a whoosh? It sounds hilarious, but I see no sign of this on youtube. 
Cookie Monster hosting “Monsterpiece Theater” and introducing the historical play 'Me, Claudius."
“Onetwothreefourfivesixseveneightniiiiiiiinetennnnnn!”
Sure thing, King!
Back before I had cable- maybe early 90’s- I was watching it before I went to work and there was a segment with Kermit the Frog singing a great version of the A-B-C’s. He was sort of roaming through a marshy swampland, and going “A-alligator; B-bayou; C-[something else]”, etc. For each letter he pointed out an animal or other thing that began with the letter, and the tune was a cool little ditty. Never saw it since but I remember thinking it was a much better version than the normal Alphabet Song jingle.
It was real. I remember this episode- it was from a couple years back. At the end, one of the humans (Gordon?) comments something to the effect of “Wow, they are a very odd couple,” to which Telly responds, “I don’t get it.” (Sesame Street is known for its pop-culture parodies- one interesting one I read online that I didn’t get a chance to see from recent years is The Count watching his favorite TV show, 24, complete with a Muppet resembling Kiefer Sutherland. He likes to count the seconds, of course. It’s amazing how the writers are able to combine parodies with educational lessons- one of the spoofs announced for this year is A’s Anatomy.)
Ernie singing “I Don’t Want to Live on the Moon”.
There was one I remember from when I was a little kid, but haven’t been able to find on YouTube. I’m not sure if it was on Sesame Street or the Muppets. What I remember was a backdrop of some kid muppets in a schoolhouse, sing-songing, “Two and two are four, four and four are eigh, eight and eight are sixteen, sixteen and sixteen are thirty-two.” Over this, a live person sang “The Garden Song.” I think an inchworm muppet was involved in some way. It was really beautiful, but I haven’t seen it since I was maybe 6, so I could be misremembering.
Pretty much anything involving Cookie Monster.
The Elmo’s World parody Cookie’s World stands out (“Confidentially, me no think you’re going to be around much longer…”), as do Monsterpiece Theater ("…with your host, me, Alistair Cookie") and the Letter Of The Day bits.
ETA – Oh yeah…Liam Nissen as The Count kicked some pretty good butt as well.
Near;
Grover running away from the camera…
FAR!..
Grover running towards the camera…
near;
Grover running away for the camera…
etc…