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) $\psi \left(x\right)=(x,y_1\left(x\right)-y_2\left(x\right))$, $x\in [0,\alpha ]$, where $y_1$, $y_2$ are convex and Lipschitz on $[0,\alpha ]$. In other words: singularities propagate along arcs with finite turn.

In Albano-Cannarsa [1] the authors proved that, under some conditions, the singularities of the semiconcave viscosity solutions of the Hamilton-Jacobi equation propagate along generalized characteristics. In this note we will provide a simple proof of this interesting result.