  We consider singular-degenerate, multivalued stochastic fast diffusion
equations with multiplicative Lipschitz continuous noise. In particular, this
includes the stochastic sign fast diffusion equation arising from the
Bak-Tang-Wiesenfeld model for self-organized criticality. A well-posedness
framework based on stochastic variational inequalities (SVI) is developed,
characterizing solutions to the stochastic sign fast diffusion equation,
previously obtained in a limiting sense only. Aside from generalizing the SVI
approach to stochastic fast diffusion equations we develop a new proof of
well-posedness, applicable to general diffusion coefficients. In case of linear
multiplicative noise, we prove the existence of (generalized) strong solutions,
which entails higher regularity properties of solutions than previously known.
