  We study a class of ordinary differential equations with a non-Lipschitz
point singularity, which admit non-unique solutions through this point. As a
selection criterion, we introduce stochastic regularizations depending on the
parameter $\nu$: the regularized dynamics is globally defined for each $\nu >
0$, and the original singular system is recovered in the limit of vanishing
$\nu$. We prove that this limit yields a unique statistical solution
independent of regularization, when the deterministic system possesses certain
chaotic properties. In this case, solutions become spontaneously stochastic
after passing through the singularity: they are selected randomly with an
intrinsic probability distribution.
