  It is proved that the solutions to the singular stochastic $p$-Laplace
equation, $p\in (1,2)$ and the solutions to the stochastic fast diffusion
equation with nonlinearity parameter $r\in (0,1)$ on a bounded open domain
$\Lambda\subset\R^d$ with Dirichlet boundary conditions are continuous in mean,
uniformly in time, with respect to the parameters $p$ and $r$ respectively (in
the Hilbert spaces $L^2(\Lambda)$, $H^{-1}(\Lambda)$ respectively). The highly
singular limit case $p=1$ is treated with the help of stochastic evolution
variational inequalities, where $\mathbbm{P}$-a.s. convergence, uniformly in
time, is established.
  It is shown that the associated unique invariant measures of the ergodic
semigroups converge in the weak sense (of probability measures).
