  We investigate the singularities of the trace of the half-wave group,
$\mathrm{Tr} \, e^{-it\sqrt\Delta}$, on Euclidean surfaces with conical
singularities $(X,g)$. We compute the leading-order singularity associated to
periodic orbits with successive degenerate diffractions. This result extends
the previous work of the third author \cite{Hil} and the two-dimensional case
of the work of the first author and Wunsch \cite{ForWun} as well as the seminal
result of Duistermaat and Guillemin \cite{DuiGui} in the smooth setting. As an
intermediate step, we identify the wave propagators on $X$ as singular Fourier
integral operators associated to intersecting Lagrangian submanifolds,
originally developed by Melrose and Uhlmann \cite{MelUhl}.
