  We prove uniform hyperbolicity of the renormalization operator for all
possible real combinatorial types. We derive from it that the set of infinitely
renormalizable parameter values in the real quadratic family $P_c: x\mapsto
x^2+c$ has zero measure. This yields the statement in the title (where
``regular'' means to have an attracting cycle and ``stochastic'' means to have
an absolutely continuous invariant measure). An application to the MLC problem
is given.

Dynamical Systems