Here we go, "grunge style":

The well known ones:

Smells Like Teen Spirit - Nivana
Jeremy - Pearl Jam
Would? - Alice In Chains
Black Hole Sun - Soundgarden

Others:
Touch Me I'm Sick - Mudhoney (definitive "grunge" song)

My favorites:
Nearly Lost You - Screaming Trees
Troubled Times - Screaming Trees
Alice Said - Screaming Trees
All I Know - Screaming Trees
Sworn and Broken - Screaming Trees
Dying Days - Screaming Trees (about the deaths of Seattle musicians)
Again - Alice In Chains
Shame In You - Alice In Chains
The Day I Tried to Live - Soundgarden
Like Suicide (original or acoustic) - Soundgarden
Blow Up the Outside World - Soundgarden
Burden in My Hand - Soundgarden
Dusty - Soundgarden
Jesus Christ Pose - Soundgarden
Room a Thousand Years Wide - Soundgarden

Some of those were thrown in because they were more popular, but they're great songs too.

This is a small list, just the biggest and my favorite bands. Check out AMG for more.

ps-I disagree with AMG, which lists Bush, Silverchair, PotUSA, and some others as "Key Artists" in grunge.