On the Morse–Sard property and level sets of Sobolev and BV functions

We establish Luzin $N$ and Morse–Sard properties for $\mathrm{BV}_2$ functions defined on open domains in the plane. Using these results we prove that almost all level sets are finite disjoint unions of Lipschitz arcs whose tangent vectors are of bounded variation. In the case of $\mathrm{W}^{2,1}$ functions we strengthen the conclusion and show that almost all level sets are finite disjoint unions of $\mathrm{C}^1$ arcs whose tangent vectors are absolutely continuous along these arcs.