My personal favourite of the 'main' Muppet gang, Gonzo, looked best to my eyes in season 2. He continued to look good in seasons 3 to 5 and up until the early 90s, but the current version of the puppet looks a little 'off'. Not sure exactly what it is... maybe the larger nose.
As for my favourite Sesame Street character - Big Bird - he looked best in the late 70s/early 80s, which would have been 'my' era. Looks a bit too 'fluffy' today, but maybe that's down to me being sentimental about my childhood.
Most people have listed their favourite versions of the main characters, so I'll concentrate on a few of my favs - mainly the full bodied Muppets.
Sweetums - Seasons 1 to 3, although he still looks good today.
Doglion - He was always a fantastic looking monster but I think he looked his best in the Cloris Leachman and Lesley Ann Warren episodes. He seems to have lost the darker brown tint to his facial fur by this point ( although he looks brown again in 'The Muppet Movie' ). He looked a bit ragged by the late 80s/early 90s, but looked great again on 'Muppets Tonight'. I think they gave him a wash & blowdry.
Thog - Don't think he changed much over the course of the 70s, so I'll just say he looked his best before the suit decayed
Looks really nice in the Kaye Ballard ep.
Fletcher Bird - Always looked pretty good, but his colors seemed really vibrant in the 1993 Royal Variety performance ( this could have been down to lighting ), plus he sported a very nice feather arrangement atop his head.
Timmy Monster - Looked great after he grew a tail in the later half of season 2. The first season version of him looks a little weird.
Mean Mamma - Oddly enough, I prefer the puppet version to the full bodied suit. Still a very nice monster overall, though.
The Mutations - They didn't really change much, but I really liked their appearance in the Madeline Kahn episode.
Behemoth, much like Mean Mamma, looked nicer as a puppet.