  We construct cosmological spacetimes with null Kasner-like singularities as
purely gravitational solutions with no other background fields turned on. These
can be recast as anisotropic plane-wave spacetimes by coordinate
transformations. We analyse string quantization to find the spectrum of string
modes in these backgrounds. The classical string modes can be solved for
exactly in these time-dependent backgrounds, which enables a detailed study of
the near singularity string spectrum, (time-dependent) oscillator masses and
wavefunctions. We find that for low lying string modes(finite oscillation
number), the classical near-singularity string mode functions are non-divergent
for various families of singularities. Furthermore, for any infinitesimal
regularization of the vicinity of the singularity, we find a tower of string
modes of ultra-high oscillation number which propagate essentially freely in
the background. The resulting picture suggests that string interactions are
non-negligible near the singularity.
