We study the set $H$ of formal arcs of a germ of a quasi-ordinary hypersurface singularity
or an affine toric variety. For $s 
\geq 0$ the set
$j^s(H)$  of $s$-jets of arcs in $H$ is a constructible subset of an affine space.
Let us denote $[ j^s(H) ]$ its image in the Grothendieck ring of varieties.
We obtain an explicit description of $[ j^s(H) ]$ in terms of some monomial ideals
defined from singularity invariants. A formula for the Denef-Loeser geometric Poincaré series is obtained,
generalizing previous results of Lejeune and Reguera in the case of normal toric surfaces.
This is a joint work in progress with H. Cobo.