  Semiclassical asymptotics for linear Schr\"odinger equations with non-smooth
potentials give rise to ill-posed formal semiclassical limits. These problems
have attracted a lot of attention in the last few years, as a proxy for the
treatment of eigenvalue crossings, i.e. general systems. It has recently been
shown that the semiclassical limit for conical singularities is in fact
well-posed, as long as the Wigner measure (WM) stays away from singular saddle
points. In this work we develop a family of refined semiclassical estimates,
and use them to derive regularized transport equations for saddle points with
infinite Lyapunov exponents, extending the aforementioned recent results. In
the process we answer a related question posed by P. L. Lions and T. Paul in
1993. If we consider more singular potentials, our rigorous estimates break
down. To investigate whether conical saddle points, such as $-|x|$, admit a
regularized transport asymptotic approximation, we employ a numerical solver
based on posteriori error control. Thus rigorous upper bounds for the
asymptotic error in concrete problems are generated. In particular, specific
phenomena which render invalid any regularized transport for $-|x|$ are
identified and quantified. In that sense our rigorous results are sharp.
Finally, we use our findings to formulate a precise conjecture for the
condition under which conical saddle points admit a regularized transport
solution for the WM.
