  Several aspects of regularity theory for parabolic systems are investigated
under the effect of random perturbations. The deterministic theory, when strict
parabolicity is assumed, presents both classes of systems where all weak
solutions are in fact more regular, and examples of systems with weak solutions
which develop singularities in finite time. Our main result is the extension of
a regularity result due to Kalita to the stochastic case. Concerning the
examples with singular solutions (outside the setting of Kalita's regularity
result), we do not know whether stochastic noise may prevent the emergence of
singularities, as it happens for easier PDEs. We can only prove that, for a
linear stochastic parabolic system with coefficients outside the previous
regularity theory, the expected value of the solution is not singular.
