I like Kermit's version, natch. But I'm afraid I'm going to have to go against the popular vote and say that it's not my favorite version.
My favorite version of Rainbow Connection is the one sung by Vonda Shepard in the Ally McBeal episode "Angels and Blimps". The episode was about a boy (Haley Joel Osment) dying of cancer who decided to sue God, and had his case represented by Ling. Osment's character dies at the end of the episode, and the second and third verses play over scenes of Ling crying in the ambulance bay of the hospital (after she coldly tells Ally to "get over" the boy's death) and an angel leading the spirit of the boy through the hospital up to Heaven.
Unfortunately only a couple chunks of the song are heard in the episode, and I wish that Vonda would have released a complete version of it on one of the Ally soundtracks. But despite those flaws, the sentiment of the song, the emotion of the episode and Vonda's voice make it my favorite rendition.
-Kim