  Families of jets through singularities of algebraic varieties are here
studied in relation to the families of arcs originally studied by Nash. After
proving a general result relating them, we look at normal locally complete
intersection varieties with rational singularities and focus on a class of
singularities we call "higher Du Val singularities", a higher dimensional (and
codimensional) version of Du Val singularities that is closely related to
Arnold singularities. More generally, we introduce the notion of "higher
compound Du Val singularities", whose definition parallels that of compound Du
Val singularities. For such singularities, we prove that there exists a
one-to-one correspondence between families of arcs and families of jets of
sufficiently high order through the singularities. In dimension two, the result
partially recovers a theorem of Mourtada on the jet schemes of Du Val
singularities. As an application, we give a solution of the Nash problem for
higher Du Val singularities.
