  We consider the Weil--Petersson (WP) metric on the modular surface. We lift
WP geodesics to the universal cover of the modular surface and analyse
geometric properties of the lifts as paths in the hyperbolic metric on the
universal cover. For any pair of distinct points in the thick part of the
universal cover, we prove that the WP and hyperbolic geodesic segments that
connect the pair, fellow-travel in the thick part and all deviations between
these segments arise during cusp excursions. Furthermore, we give a
quantitative analysis of the deviation during an excursion.
  We leverage the fellow traveling to derive a correspondence between recurrent
WP and hyperbolic geodesic rays from a base-point. We show that the
correspondence can be promoted to a homeomorphism on the circle of directions.
By analysing cuspidal winding of a typical WP geodesic ray, we show that the
homeomorphism pushes forward a Lebesgue measure on the circle to a singular
measure. In terms of continued fraction coefficients, the singularity boils
down to a comparison that we prove, namely, the average coefficient is bounded
along a typical WP ray but unbounded along a typical hyperbolic ray.
