  We study the principal configurations around an isolated $\eta$-umbilical
point on a generic spacelike surface $S$ immersed in a null hypersurface $M$ of
Minkowski space $\mathbb{R}_1^4$ relative to a well-defined null vector field
$\eta$ orthogonal to the surface $S$. In the particular case of $M$ being a
null rotation hypersurface of $\mathbb{R}_1^4$ we also recover the local
Darbouxian principal configurations around that kind of $\eta$-umbilical
points.
