  We study Malliavin differentiability of solutions to sub-critical singular
parabolic stochastic partial differential equations (SPDEs) and we prove the
existence of densities for a class of singular SPDEs. Both of these results are
implemented in the setting of regularity structures. For this we construct
renormalized models in situations where some of the driving noises are replaced
by deterministic Cameron-Martin functions, and we show Lipschitz continuity of
these models with respect to the Cameron-Martin norm. In particular, in many
interesting situations we obtain a convergence and stability result for lifts
of $L^2$-functions to models, which is of independent interest. The proof also
involves two separate algebraic extensions of the regularity structure which
are carried out in rather large generality.
