  We study isometric cohomogeneity one actions on the (n+1)-dimensional
Minkowski space up to orbit-equivalence. We give examples of isometric
cohomogeneity one actions on the Minkowski space whose orbit spaces are
non-Hausdorff. We show that there exist isometric cohomogeneity one actions on
the Minkowski space which are orbit-equivalent on the complement of an
n-dimensional degenerate subspace and not orbit-equivalent on this degenerate
subspace. We classify isometric cohomogeneity one actions on 2- and
3-dimensional Minkowski spaces up to orbit-equivalence.
