  Higher rational and higher Du Bois singularities have recently been
introduced as natural generalizations of the standard definitions of rational
and Du Bois singularities. In this note, we discuss these properties for
isolated singularities, especially in the locally complete intersection (lci)
case. First, we reprove the fact that a $k$-rational isolated singularity is
$k$-Du Bois without any lci assumption. For isolated lci singularities, we give
a complete characterization of the $k$-Du Bois and $k$-rational singularities
in terms of standard invariants of singularities. In particular, we show that
$k$-Du Bois singularities are $(k-1)$-rational for isolated lci singularities.
In the course of the proof, we establish some new relations between invariants
of isolated lci singularities and show that many of these vanish. The methods
also lead to a quick proof of an inversion of adjunction theorem in the
isolated lci case. Finally, we discuss some results specific to the
hypersurface case.
