This was a tough one. (Although, all things being equal, if the toughest part of your day is deciding which version of "Rainbow Connection" you like the best, you're having a better day than 80 per cent of the planet. But I digress...)
After giving it a bit of thought, I've ranked the original TMM recording with Jim Henson as Kermit on top, but only slightly ahead of my close second (from The Muppets) and third (Weezer and Hayley Williams from The Green Album).
The original is still my favourite because I feel Jim's/Kermit's vocals best capture the song's hopeful optimism. There's an organic feel to the musical arrangement - especially the banjo, which (appropriately enough) never abandons its role as the backbone of the piece - and we never doubt for a moment that it's being sung by a frog sitting on a log in a swamp nestled in a lush green forest. And, as far as the movie goes, the scene is the perfect opener for the first full-length Muppet film. They're not in the screening room or a studio setting anymore; this is real - and the song beautifully conveys real emotions that we have all felt at some point in our lives.
(I must admit to some bias because my parents bought me the TMM soundtrack in the fall of 1980, after I had spent my eighth birthday in a hospital having eye surgery, and Jim's/Kermit's calm delivery was just what I needed during the recovery period. The recording is also a favourite of an uncle of mine that hosted us at his place during this time.)
That being said, I was blown away by the new version recorded for last year's release of The Muppets. When I first heard "Rainbow Connection" was being revived for the new movie, I had a mixture of anticipation and scepticism and wondered if Segel and Stoller were going overboard on the nostalgia angle. But oh, how good it felt to hear that banjo plucking again as the curtains opened.
It was lovingly reworked as an ensemble number, both in the movie's edited version and on the soundtrack. Eric's performance as Piggy was a delightful surprise (I never considered this a Piggy song) and a marvelous counterbalance to Steve's faithful recreation of the song. From a pure visual standpoint, it was beautifully shot, and that last pull-back from Animal's return to the drums is already one of my favourite Muppet moments of all time. I can't watch or listen to it without picturing Jim Henson and Walt Disney shaking hands in heaven.
And the Weezer/Hayley Williams cover works for me for three key reasons: (1) Weezer's Muppet-cred from the "Keep Fishing" video, (2) the creative-yet-faithful instrumentation (particularly the beginning and ending harp lines and the little sound effects that recreate that "out in the swamp" vibe), and (3) the childlike innocence that Rivers Cuomo and Hayley Williams bring to their vocals. You can picture them being eight or nine years old and singing it the same way.
Very fond of the Sarah McLachlan and Willie Nelson covers too, as well as Kermit's TMS duet with Debby Harry (I never thought she'd do a song like that) and his pre-Oscar duet with Darren Criss. I must admit I haven't heard most of the others - I'll have to go check them out. Thanks for putting up a great poll!