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.