  Motivated by recent experiments on possible supersolid behavior of $^4$He
solids at low temperature, we consider a model of superfluidity in a defected
solid containing a system spaning network of correlated linear dislocations, or
planar grain boundaries. Using arguments based on the Harris criterion, as well
as numerical simulations, we find that such correlated quenched disorder shifts
the familiar superfluid lambda transition to a new disordered universality
class in which the correlation length exponent $\nu\ge 1$. This results in the
temperature-derivates for the superfluid density, $d\rho_{\rm s}/dT$, and for
the heat capacity, $dc/dT$, remaining finite at the transition $T_{\rm c}$, and
thus a less singular transition, profoundly different from the usual lambda
transition.
