  P. Albano and P. Cannarsa proved in 1999 that, under some applicable
conditions, singularities of semiconcave functions in $\R^n$ propagate along
Lipschitz arcs. Further regularity properties of these arcs were proved by P.
Cannarsa and Y. Yu in 2009. We prove that, for $n=2$, these arcs are very
regular: they can be found in the form (in a suitable Cartesian coordinate
system) $\psi(x) = (x, y_1(x)-y_2(x)), x \in [0,\alpha]$, where $y_1$, $y_2$
are convex and Lipschitz on $[0,\alpha]$. In other words: singularities
propagate along arcs with finite turn.
