Are you asking for performances of songs by The Muppets or songs unique to The Muppets? Or maybe both? Because The Muppets did lots of great songs on The Muppet Show- but of course many (if not most) of them were first made popular by various artists and Broadway shows, etc. Obviously The Muppet Show theme song is original and Gonzo's song in the Paul Simon episode is original, but I'm not sure how many songs beyond that were written specifically for the show.
Regardless, some of my favorites (and some of which I only really know from The Muppets) would include
The Muppet Show theme song
New York State of Mind (Billy Joel)
Crocodile Rock
Don't Go Breaking My Heart
Bennie and the Jets
Goodbye Yellow Brick Road (last 4 by Elton John)
In the Navy (The Village People)
Going to Rio
Lullaby of Broadway
Halfway Down the Stairs
The Lonely Goatherd (from The Sound of Music)
One Way or Another
Call Me (last 2 by Blondie)
Welcome to My Nightmare
You and Me
School's Out (last 3 by Alice Cooper)
Scarborough Fair
It's Been a Long, Long Day
50 Ways to Leave Your Lover
Loves Me Like a Rock (last 4 by Paul Simon)
Put The Lime in the Coconut
The Banana Boat Song (Day-O)
Turn the World Around (last 2 by Harry Belafonte)
Grandma's Feather Bed
It's in Every One of Us (last 2 performed by John Denver)
You and Me and George
Jubilee
Here You Come Again
Love Will Keep Us Together (last 2 performed by Leslie Uggams)
You Can't Roller Skate in a Buffalo Herd
We Must Believe in Magic (as performed by Crystal Gayle)
You Make Me Feel Like Dancing
When I Need You (last 2 by Leo Sayer)
Singing in the Rain medley by Gene Kelly
You've Got a Friend (James Taylor song performed by Vincent Price)
Mahna Mahna
Inchworm
Time In a Bottle (great Jim Croce song)
Rocking Robin
Staying Alive (by The Bee Gees)
Ethel Merman's medley of Broadway songs
and definitely
Just One Person
and probably tons others I forgot. But those are some good ones right there anyway.