It is well known that when gravitational plane waves propagating and colliding in an otherwise flat background interact they produce singularities. In this paper we explore the structure of the singularities produced in the collisions of arbitrarily polarized gravitational plane waves and we consider the problem of whether (or under what conditions) singularities can be produced in the collisions of almost-plane gravitational waves with finite but very large transverse sizes. First we analyze the asymptotic structure of a general arbitrarily polarized colliding plane-wave spacetime near its singularity. We show that the metric is asymptotic to a generalized inhomogeneous-Kasner solution as the singularity is approached. In general, the asymptotic Kasner axes as well as the asymptotic Kasner exponents along the singularity are functions of the spatial coordinate that runs tangentially to the singularity in the non-plane-symmetric direction.