Question
I have some type of small moles in my yard. These do not burrow underground. there are no raised tunnels - rather there are sqiggly trails above the ground, mainly under my shade tree. I have killed a couple of these "moles" they are about the size of mouse and charcoal gray in color.

Answer
I suspect that your problem is not moles, but voles. Voles makes about ground "runway" tracks. They look like gray mice.

See descriptions below:
http://www.urbanext.uiuc.edu/macon/inout/020317.html

I also have a link with suggested remedies:
http://www.ipm.ucdavis.edu/PMG/PESTNOTES/pn7439.html