  In this paper linear stochastic transport and continuity equations with drift
in critical $L^{p}$ spaces are considered. In this situation noise prevents
shocks for the transport equation and singularities in the density for the
continuity equation, starting from smooth initial conditions. Specifically, we
first prove a result of Sobolev regularity of solutions, which is false for the
corresponding deterministic equation. The technique needed to reach the
critical case is new and based on parabolic equations satisfied by moments of
first derivatives of the solution, opposite to previous works based on
stochastic flows. The approach extends to higher order derivatives under more
regularity of the drift term. By a duality approach, these regularity results
are then applied to prove uniqueness of weak solutions to linear stochastic
continuity and transport equations and certain well-posedness results for the
associated stochastic differential equation (sDE) (roughly speaking, existence
and uniqueness of flows and their $C^\alpha$ regularity, strong uniqueness for
the sDE when the initial datum has diffuse law). Finally, we show two types of
examples: on the one hand, we present well-posed sDEs, when the corresponding
ODEs are ill-posed, and on the other hand, we give a counterexample in the
supercritical case.
