The tools I like to use

Most sites I have visited have suggested either small sharp scissors or an exacto knife.  I find I can get more detail with an exacto knife.  I discovered the swivel exacto knife that fits around my finger for comfort.  I love the swivel!  It is a very tiny blade and not as easy to find the replacement blades, but it works so well on curves.

I also use the other common tools like a self healing mat for cutting on, and acid free glue stick for adhering the image onto the background paper.  BUT!  I found a self healing mat that rotates!  LOVE IT!  Between the swivel knife and rotating mat, the images almost cut themselves!  I also bought an awl for poking tiny holes for certain effect.  But I use a folded paper towel between the image and the mat as I poke otherwise the holes are to tiny to notice once mounted.
I buy my paper which is black on one side and white on the other on line.  Amazon has some good books for Scherenschnitte that have designs ready to cut. 
Of course my last essential tool is my cheater eye glasses which magnify everything so each cut is exact, I hope!  Warning: I have noticed my eyeglasses I use for reading don't seem to be strong enough anymore.  Not sure if the magnifiers are too strong or I'm simply showing my age :(  

The exacto knives come in so many varieties, but I really like the one that wraps around the finger, especially with my arthritis.

These mats can be expensive but find a coupon and it's worth it.  After some time my little mat got too many groves cut in that could no longer "heal", I had to get another one. 

Don't skimp on your tools.  Replace the blades and your work will show it.