  The singular set of a viscosity solution to a Hamilton-Jacobi equation is
known to propagate, from any noncritical singular point, along singular
characteristics which are curves satisfying certain differential inclusions. In
the literature, different notions of singular characteristics were introduced.
However, a general uniqueness criterion for singular characteristics, not
restricted to mechanical systems or problems in one space dimension, is missing
at the moment. In this paper, we prove that, for a Tonelli Hamiltonian on
$\mathbb{R}^2$, two different notions of singular characteristics coincide up
to a bi-Lipschitz reparameterization. As a significant consequence, we obtain a
uniqueness result for the class of singular characteristics that was introduced
by Khanin and Sobolevski in the paper [On dynamics of Lagrangian trajectories
for Hamilton-Jacobi equations. Arch. Ration. Mech. Anal., 219(2):861-885,
2016].
