  We provide a smoothening criterion for group actions on manifolds by singular
diffeomorphisms. We prove that if a countable group $\Gamma$ has the fixed
point property FW for walls (e.g. if it has property (T)), every aperiodic
action of $\Gamma$ by diffeomorphisms that are of class $C^r$ with countably
many singularities is conjugate to an action by true diffeomorphisms of class
$C^r$ on a homeomorphic (possibly non-diffeomorphic) manifold. As applications,
we show that Navas's result for actions of Kazhdan groups on the circle, as
well as the recent solutions to Zimmer's conjecture, generalise to aperiodic
actions by diffeomorphisms with countably many singularities.
