A lot of my Muppet fascination tends to swing towards background or rarely used characters. Not sure why this is - maybe said characters' lack of screentime makes their appearances all the more special. It's wonderful that such detail and care was put into even minor characters, something that displays perfectly the skill and perfection of Henson and his team.
A few people have picked Boppity and he'd be high on my list too. A really nicely designed monster who appeared quite a lot but was mainly confined to crowd scenes or sharing the screen with other, more well known characters. I really like him, though, and would love to own a replica. People who've read my posts will also know that I'm a big fan of Fletcherbird ( lovely design and beautifully performed ), but as he was the main Muppet in at least two songs and had very prominent roles in a couple of others, I think he falls just slightly outside the definition of 'background character'. The Gawky Birds, characters I also regard as fascinating, are more in line with the definition. They never spoke but looked absolutely fantastic, particularly in the Bruce Forsyth episode. Actually, I clearly have a thing for Muppet birds as I also love the Whaddayasay Bird, plus the others that appeared in that same sketch. And then there's the duck who turned up in a few episodes too. And let's not forget the penguins. To be honest, I think pretty much every minor character had its merits, from the Javas ( quite brilliant ) to the Hugga Wugga ( hypnotically excellent ) to The Luncheon Counter Monster ( quite... hungry ).
The main Muppet gang were all brilliant, but equal care and attention was given to the the wonderful creatures who hid in the background and fleshed out their world. That's what makes 'The Muppet Show' a genuine television classic.