  We further investigate the properties of an approach to topological
singularities through free discontinuity functionals of Mumford-Shah type
proposed in \cite{DLSVG}. We prove the variational equivalence between such
energies, Ginzburg-Landau, and Core-Radius for anti-plane screw dislocations
energies in dimension two, in the relevant energetic regimes $|\log
\varepsilon|^a$, $a\geq 1$, where $\varepsilon$ denotes the linear size of the
process zone near the defects.
  Further, we remove the \emph{a priori} restrictive assumptions that the
approximating order parameters have compact jump set. This is obtained by
proving a new density result for $\mathbb S^1$-valued $SBV^p$ functions,
approximated through functions with essentially closed jump set, in the strong
$BV$ norm.
