  This article introduces the notion of L-tangle-free compact hyperbolic
surfaces, inspired by the identically named property for regular graphs. Random
surfaces of genus g, picked with the Weil-Petersson probability measure, are (a
log g)-tangle-free for any a < 1. This is almost optimal, for any surface is (4
log g + O(1))-tangled. We establish various geometric consequences of the
tangle-free hypothesis at a scale L, amongst which the fact that closed
geodesics of length < L/4 are simple, disjoint and embedded in disjoint
hyperbolic cylinders of width $\ge$ L/4.
