The singularity structure of maximally extended colliding plane gravitational wave spacetimes is studied. Three kinds of colliding plane waves are investigated: the impulsive (i.e. \(\delta\)-function) waves, sandwich waves and thick waves. Applying the Ellis-Schmidt [cf. G. F. R. Ellis and B. G. Schmidt, Gen. Relativ. Gravitation 8, 915- 953 (1977; Zbl 0434.53048)] classification of spacetime singularities it is found that besides the well known scalar curvature singularities in the future of the collision there exist also singularities of the other two types forming the boundary. Approximate computer calculations (program SHEEP) show that collisions of the impulsive and sandwich planes give rise to mild, topological singularities - the quasi-regular ones, while colliding thick waves result in non-scalar curvature singularities. For the case of colliding impulsive waves it is found that test particles are focused onto the quasi-regular singularity and the stress tensor of the ordinary scalar field is divergent there, indicating that physically the singularity is converted into a scalar curvature one. It is argued that in the other two cases the singularities are also unstable.