  Motivated by Hilbert's 16th problem we discuss the probabilities of
topological features of a system of random homogeneous polynomials. The
distribution for the polynomials is the Kostlan distribution. The topological
features we consider are type-$W$ singular loci. This is a term that we
introduce and that is defined by a list of equalities and inequalities on the
derivatives of the polynomials. In technical terms a type-$W$ singular locus is
the set of points where the jet of the Kostlan polynomials belongs to a
semialgebraic subset $W$ of the jet space, which we require to be invariant
under orthogonal change of variables. For instance, the zero set of polynomial
functions or the set of critical points fall under this definition.
  We will show that, with overwhelming probability, the type-$W$ singular locus
of a Kostlan polynomial is ambient isotopic to that of a polynomial of lower
degree. As a crucial result, this implies that complicated topological
configurations are rare. Our results extend earlier results from Diatta and
Lerario who considered the special case of the zero set of a single polynomial.
Furthermore, for a given polynomial function $p$ we provide a deterministic
bound for the radius of the ball in the space of differentiable functions with
center $p$, in which the $W$-singularity structure is constant.
