  We consider stationary solutions of the three dimensional Navier--Stokes
equations (NS3D) with periodic boundary conditions and driven by an external
force which might have a deterministic and a random part. The random part of
the force is white in time and very smooth in space. We investigate smoothness
properties in space of the stationary solutions. Classical technics for
studying smoothness of stochastic PDEs do not seem to apply since global
existence of strong solutions is not known. We use the Kolmogorov operator and
Galerkin approximations. We first assume that the noise has spatial regularity
of order $p$ in the $L^2$ based Sobolev spaces, in other words that its paths
are in $H^p$. Then we prove that at each fixed time the law of the stationary
solutions is supported by $H^{p+1}$. Then, using a totally different technic,
we prove that if the noise has Gevrey regularity then at each fixed time, the
law of a stationary solution is supported by a Gevrey space. Some informations
on the Kolmogorov dissipation scale are deduced.
