Plane-Symmetric Vacuum Solutions with Null Singularities for Inhomogeneous Models and Colliding Gravitational Waves

New exact vacuum solutions with various singularities in the plane-symmetric spacetime are shown, and they are applied to the analysis of inhomogeneous cosmological models and colliding gravitational waves. One of the singularities can be true null singularities, whose existence was locally clarified by Ori. These solutions may be interesting from the viewpoint of the variety of cosmological singularities and the instability problem of Cauchy horizons inside black holes.