What's the evidence that the Sumerians and Babylonians used base 60 for ordinary counting?

Good question. Since Sumerian and Akkadian are no longer spoken, we only have written records to go by, and that already preselects what kind of counting was getting done. (I'll rummage through some of my reference books and see what's to be seen.) Not sure if anybody knows how the common counting folk did it. One might investigate the words for numbers and seeing if any clues could be had from them.

Ceci n'est pas un seing.