Can anyone recommend a great way to cut perfect circles out of sticky backed velvet or vinyl for pupils? I've tried using a punch scribe set I bought online, and it doesn't work. I've been wanting a clean edge that looks great and perfectly round and I'm stuck!

What about using a stencil and a scalpel/box cutter?  I have to admit that I'm otherwise stumped; the stickiness will prevent punches from working very well. You could always just avoid using that particular material and work with something that's pre-cut into circles, like stickers or foam dots (the actual name escapes me...).

Have you tried the Olfa Compass Cutters http://www.olfa.com/CircleCuttersDetail.aspx?C=51&Id=65 I've used these on felt and fleece, and foamies.
fiskars also has some good punches I've used on plastic, but then you'll always get the same size.
I tried using and exacto around coins too.

Have you tried the Olfa Compass Cutters http://www.olfa.com/CircleCuttersDetail.aspx?C=51&Id=65 I've used these on felt and fleece, and foamies.
fiskars also has some good punches I've used on plastic, but then you'll always get the same size.
I tried using and exacto around coins too.

Oooh... I didn't know those existed. (Damned Americans - why do you get all the cool tools? ) That'd be great for cutting out my shadow puppets. I always have a really hard time cutting out circles. Have you ever used the compass cutter on plastic?

Do you have a cutting mat?  The two things I've found when cutting out stuff like this that is important, is a sharp blade and a good cutting mat.

This works on sticky back vinyl. http://shop.hobbylobby.com/1-circle-craft-lever-punch/

I'll try it on e sticky back craft foam and report back.

The one thing about the sticky back vinyl is if you are planning on using it on the rounded half dome eyes it doesn't lay very nicely.  The get the pupil large enough the vinyl wrinkles.

http://www.creativemin.com/category/foamwerks
These are designed for foamboard, but might be worth a try. 

Update: the 1" circle punch works great on 2mm sticky back craft foam. 3mm is to thick.

It does NOT work on fabric or tape.

Thank you so much! These are all great and wonderful suggestions! I will look into them and let you know how it goes!!!

I don't cut them at all. I go to Walmart, and get sticky back circles that are made to go on the bottom of what-have-you knick-knack kinds of things, to keep them from scratching your nice smooth furniture. They come in all kinds of sizes, in one little pack.

My Dad loaned me a metal pipe piece that was sharpened on on end with a file and we cut a stack of card stock circles by hitting it with a rubber hammer over piece of wood.  We were making a ton of golden coins for treasure chest and wanted a lot fast, it worked great.  I use that technique with foam sheets to get a smooth edge. For sticky backed cutting I would wipe any cutter with alcohol to remove the sticky residue after.  I made the mistake of sewing over sticky back and my needle got gummy.  For large pupil I have cut a small dart in the circle to get it to lay flat.  Have not tried this but if it is strechy maybe you could form it with heat and streching over  a ball shape before cutting or remove backing.