I shred almost everything.
If it is a bill or any other type of communication to me, I shred it. The only thing I do not shred is junk mail catalogs and flyers.
Sometimes if I purchase something and I don't want the neighbors, garbage man or the guys at the recycling center to know about it, I'll shred the package and instructions that came with it.
Example: [SHREDDED]
Sorry... I even shredded that.
When it's time to put out my shredded paper, I'll put it at the bottom of the garbage bag and then dump the contents of my cat's litter box on top of it. Have fun digging through that!
Sure, some may say I'm paranoid, but I have NEVER had anyone steal any of my personal information.