I will describe how basic ergodic theory can be used to prove that certain amenable groups are stable. I will demonstrate our method by showing that lamplighter groups are stable. Another uncountably infinite family to which our method applies are the so-called B.H. Neumann groups. This is the first construction of an uncountable family of stable groups. The talk is based on joint work with Alex Lubotzky, relying on the Invariant-Random-Subgroup criterion for stability developed by Becker-Lubotzky-Thom.