  We present regularity results for the crack set of a minimizer for the
Griffith fracture energy, arising in the variational modeling of brittle
materials. In the planar setting, we prove an epsilon-regularity result showing
that the crack is locally a $C^{1,1/2}$ curve outside of a singular set of zero
Hausdorff measure. The main novelty is that, in contrast to previous results,
no topological constraints on the crack are required. Afterwards, we show that
the crack is uniformly rectifiable which in turn allows us to prove that the
singular set has dimension strictly less than 1. The results also apply to
almost-minimizers. As a byproduct, we prove higher integrability for the
gradient of local minimizers of the Griffith energy, providing a positive
answer to the analogue of De Giorgi's conjecture for the Mumford-Shah
functional.
