These days, Kellogg's cereals have Disney wobblers in them. Because of their round bottoms and a very low center of gravity, they wobble but end up standing upright. I was discussing this with Max, who had lots of questions about center of gravity. Could you make an object with a center of gravity on the ground? No, because some of the mass has to be on the other side of any line drawn through the center of gravity.
Then I remembered something I'd read about a long time ago. A few searches, and I found it: a unistable polyhedron is one which will only rest on one of its faces. Robert Dawson calls them by a different name, but has some nice pictures: monostatic polytopes.
In the real world, there's a very pragmatic reason to be concerned about polyhedral stability: it is at the heart of the issue of Fair Dice.