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Bernadette Peters and the Muppets singing "Just One Person" to Robin, who is feeling ignored and overlooked. It's a beautiful moment, but the Muppets managed to top it when they reprise the song, twice, after Jim Henson's death (once during the memorial service with Scooter as the lead singer, once during the Muppets Celebrate Jim Henson TV special with Robin as the lead singer), dedicating the song to him. It is a serious Tear Jerker, but also warm and heartfelt.
Robin: "You know, this Jim Henson may be gone, but maybe he's still here too, inside us — believing in us!"
Pretty much the entire The Muppets Remember Jim Henson special, in fact.
Even Statler and Waldorf couldn't find anything to say after hearing those letters. They even sang along.
When Kermit, now operated by Steve Whitmire, enters and smiles at the tribute. For Kermit to move on, that's exactly what Jim wanted. And photos of Jim Henson are visible.
The second season had two very similar sketches that were just made of d'aww: one where Rowlf sings What a Wonderful World to a puppy, and one where Link Hogthrob sings Sonny Boy to a piglet. In both cases the "children" were played by actual live animal babies, multiplying the cuteness and heartwarming factors by several orders of magnitude.
Harry Belafonte and the Muppets singing "Turn the World Around". I dare anyone to watch that and not come away singing and smiling.
The creators were apparently well aware that what they were doing was special. At the end, even Statler and Waldorf join in the singing. This was something even they couldn't snark at.
Belafonte's lead-in to the song deserves a mention too:
"All of us, we're here for a very short time, and in that time it makes no difference in any of us, if we take time to understand one another. So the question is: do you know who I am? Do I know who you are? Do we care about each other? Because if we do...then together, we can turn the world around."
Henson stated that "Turn the World Around" was his favorite segment from the show. Fittingly, Belafonte performed it at Henson's memorial service.
From the Gene Kelly episode: The final sketch, where Rowlf keeps trying to get Gene to sing "Singin' in the Rain" by playing the opening vamp. Gene is reluctant to sing it because he doesn't think he will ever be able to do it as well as in the movie, so each time he uses the vamp to transition into one of his other famous songs. Finally, Rowlf gives up, and then Gene tells him to play it again because he knows another song the vamp goes with — which is, of course, "Singin' in the Rain." This sketch is a Crowning Moment of Heartwarming in part because this was the final sketch of the final episode to be taped (though not the last one broadcast). It just seems a very moving end to the series' long run. Watch it here.
A bit more on Jim Henson, but his funeral was... unique. He had stated in his will that no one would wear black. Everyone who appeared at the funeral was handed a specially-crafted, rainbow-colored puppet butterfly they could wave along with the presentation. And during the funeral one of the songs sung was his favorite, "Lydia the Tattooed Lady". ... they also sang Rainbow Connection...
Danny Kaye wanders backstage and hears Statler and Waldorf apparently insulting him. It's saved from being a Tearjerker when Statler and Waldorf meet Kaye and are shocked to realize he's this week's guest, and not the person they were talking about, and gush over how much they love him.
Plus, Kaye's backstage performance of "Inchworm."
The entire episode, really. Perhaps the clearest it ever was just how much everyone involved with the show adored the guest.
The Muppets singing "Consider Yourself" to Edgar Bergen, a beautiful tribute to one of Jim Henson's biggest inspirations.
Any time Kermit and Piggy have a falling out and inevitably get back together in the end, especially in the Loretta Swit episode, and in The Muppets. After Kermit fires her and storms out of the scene, Piggy is initially, uncharacteristically aghast and says to herself, "...he doesn't really mean it..." That small moment really shows how much she loves him.
Marty Feldman and Cookie Monster bonding over their similar googly eyes.
After every show, the Muppets thank their guest stars with a Muppet version of them to take and keep for themselves. It just says a lot that the creators would give a sort of tribute and well-thought out gift to thank them for their time.
That was actually dropped after the first two episodes, although later on in the first season, Paul Williams received one of the Paul Williams Muppets used in, "Just an Old Fashioned Love Song", and Jason Segel receive the Jason Segel Muppet used in the 2011 film.
At the end of the Statler and Waldorf song "It Was A Very Good Year" Waldorf kind of cuddles next to Statler. It's brief and not too noticable but it shows the cute friendship between the two, despite how they bicker and argue.
Something's Missing, in which a muppet with an incomplete face sings about how he always gets the short end of the stick, only for that to change when he's around his love.
"Now my love, I beg you, stay, cause when you stay I can say, 'Nothing's missing, nothing's missing, nothing's missing!'"
In the Peter Ustinov episode Kermit is feeling overlooked and neglected by his friends over the guest star which eventually culminates in him singing "It's not Easy Being Green". While it starts out sad in tone with Kermit wanting to stand out so his friends praise him again it ends with Kermit realizing that he has to get over his jealousy and accept himself.
In the Kenny Rogers episode Kermit is injured during an act and his friends immediately stop the show to see if he's okay with Fozzie asking for a doctor in the audience. It was a very sweet way of showing that for all that they drive each other crazy that at the end of the day the Muppets are True Companions.