Can free strings propagate across plane wave singularities?

We study free string propagation in families of plane wave geometries developing
strong scale-invariant singularities in certain limits. We relate the singular limit of
the evolution for all excited string modes to that of the center-of-mass motion (the latter
existing for discrete values of the overall plane wave profile normalization). Requiring that
the entire excitation energy of the string should be finite turns out to be quite restrictive
and essentially excludes consistent propagation across the singularity, unless dimensionful
scales are introduced at the singular locus (in an otherwise scale-invariant space-time).