Natural conclusion that we can draw from this question about cutting small circles is that it is obvious the cutting tool or implement would need to be very sharp to be effective.
I used to use cemented floor for placing the paper for such papercutting when I was a child.
Since the tools would be that sharp they are going to affect the working surface below the paper. Another concern would be the paper moving around on that working medium.
Considering the above factors, what medium should be kept beneath the paper when papercutting?