Well I'd definitely choose several moments from The Muppet Show, including the following:
Robin singing Halfway Down The Stairs
Robin in the Bernadette Peters episode, particularly where he is doing Just One Person with the Guest Star.
Floyd singing While My Guitar Gentley Weeps.
whoever it was (I think it was one of Jim Henson's characters) singing Time In A Bottle
Kermit going out on stage and singing 'Bein' Green'
Floyd sining New York State Of Mind
Gonzo singing Wishing Song
From the films I would identify two really cute moments
Kermit singing Rainbow Connection in The Muppet Movie
Robin singing Bless Us All in The Muppet Christmas Carol
From the Muppet albums
One which really stands out is Kermit (performed by Steve Whitmire) singing 'Have Yourself A Merry Little Christmas' on the album 'A Green And Red Christmas'
And now, you did specifically include in the brief things that were Muppet-related, With that in mind I would like to highlight two moments which I hope could be counted as "Muppet related"
The song 'Eye Of The Storm' on Jerry Nelson's album 'Truro Daydreams'.
and, 'Rainbow Connection' as performed by West End actress Louise Gold as the encore number in her cabaret act. It's very beautiful and moving, and well cute in a moving sentimental way.