We study a random perturbation of the Euclidean plane, and show that it is unlikely that the distance-minimizing path between the two points can be extended into an infinite distance-minimizing path. More precisely, we study a large class of planar first-passage percolation models and show that the probability that a given site is visited by an infinite geodesic starting at the origin tends to zero uniformly with the distance. In particular, this show that the collection of infinite geodesics starting at the origin covers a negligible fraction of the plane. This provides the first progress on the `highways and byways' problem, posed by Hammersley and Welsh in the 1960s.