  We prove optimal regularity estimates in Sobolev spaces in time and space for
solutions to stochastic porous medium equations. The noise term considered here
is multiplicative, white in time and coloured in space. The coefficients are
assumed to be H\"older continuous and the cases of smooth coefficients of at
most linear growth as well as $\sqrt{u}$ are covered by our assumptions. The
regularity obtained is consistent with the optimal regularity derived for the
deterministic porous medium equation in [Gess 2020] and [Gess, Sauer, Tadmor
2020] and the presence of the temporal white noise. The proof relies on a
significant adaptation of velocity averaging techniques from their usual $L^1$
context to the natural $L^2$ setting of the stochastic case. We introduce a new
mixed kinetic/mild representation of solutions to quasilinear SPDE and use
$L^2$ based a priori bounds to treat the stochastic term.
