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We describe the irreducible components of the jet schemes with origin in the singular locus of a two-dimensional quasi-ordinary hypersurface singularity. A weighted graph is associated with these components and with their embedding dimensions and their codimensions in the jet schemes of the ambient space. 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 (equivalent to a divisorial valuation on
), that computes the log-canonical threshold of the singularity embedded in
. This provides us with pairs
whose log-canonical thresholds are not computed by monomial divisorial valuations. Note that for a pair
is a plane curve, the log-canonical threshold is always computed by a monomial divisorial valuation (in suitable coordinates of
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