  We prove that the solution to the singular-degenerate stochastic
fast-diffusion equation with parameter $m\in (0,1)$, with zero Dirichlet
boundary conditions on a bounded domain in any spatial dimension, and driven by
linear multiplicative Wiener noise, exhibits improved regularity in the Sobolev
space $W^{1,m+1}_0$ for initial data in $L^{2}$.
