We construct a class of plane-symmetric solutions possessing a curvatu
re singularity that is null and weak, like the space-time singularity
at the Cauchy horizon of spinning (or charged) black holes. We then an
alyze the stability of this singularity using a rigorous nonperturbati
ve method. We find that within the framework of (linearly polarized) p
lane-symmetric space-times this type of null weak singularity is local
ly stable. Generically, the singularity is also scalar curvature. Thes
e observations support the new picture of the null weak singularity in
side spinning (or charged) black holes, which is so far established pr
imarily on the perturbative approach.