  We prove partial regularity for minimizers to elasticity type energies in the
nonlinear framework {with $p$-growth, $p>1$,} in dimension $n\geq 3$. It is an
open problem in such a setting either to establish full regularity or to
provide counterexamples. In particular, we give an estimate on the Hausdorff
dimension of the potential singular set by proving that is strictly less than
{$n-(p^*\wedge 2)$, and actually $n-2$ in the autonomous case} (full regularity
is well-known in dimension $2$).
  The latter result is instrumental to establish existence for the strong
formulation of Griffith type models in brittle fracture with nonlinear
constitutive relations, accounting for damage and plasticity in space
dimensions $2$ and $3$.
