  In this paper we give a complete description of the irreducible components of
the jet schemes (with origin in the singular locus) of a two-dimensional
quasi-ordinary hypersurface singularity. We associate with these components and
with their codimensions and embedding dimensions, a weighted graph. We prove
that the data of this weighted graph is equivalent to the data of the
topological type of the singularity. We also determine a component of the jet
schemes (or equivalently, a divisor on $\mathbb{A}^3$), that computes the log
canonical threshold of the singularity embedded in $\mathbb{A}^3$. This
provides us with pairs $X\subset\mathbb{A}^3$ whose log canonical thresholds
are not contributed by monomial divisorial valuations. Note that for a pair
$C\subset\mathbb{A}^2$, where $C$ is a plane curve, the log canonical threshold
is always contributed by a monomial divisorial valuation (in suitable
coordinates of $\mathbb{A}^2$).
