  Thom polynomials of singularities express the cohomology classes dual to
singularity submanifolds. A stabilization property of Thom polynomials is known
classically, namely that trivial unfolding does not change the Thom polynomial.
In this paper we show that this is a special case of a product rule. The
product rule enables us to calculate the Thom polynomials of singularities if
we know the Thom polynomial of the product singularity. As a special case of
the product rule we define a formal power series (Thom series, Ts_Q) associated
with a commutative, complex, finite dimensional local algebra Q, such that the
Thom polynomial of {\em every} singularity with local algebra Q can be
recovered from Ts_Q.
