  In this note, we give several equivalent characterizations of higher Du Bois
and higher rational singularities in the context of globally defined
hypersurfaces. As a key input, we characterize these singularities using the
Hodge filtration on the vanishing cycle mixed Hodge module. As an application,
we indicate a homological criterion for detecting higher Du Bois and higher
rational hypersurface singularities, formulated in terms of the spectral
Hirzebruch-Milnor characteristic classes.
