  We show that the sublinearly Morse directions in the visual boundary of a
rank-1 CAT(0) space with a geometric group action are generic in several
commonly studied senses of the word, namely with respect to Patterson-Sullivan
measures and stationary measures for random walks. We deduce that the
sublinearly Morse boundary is a model of the Poisson boundary for finitely
supported random walks on groups acting geometrically on rank-1 CAT (0) spaces.
We prove an analogous result for mapping class group actions on Teichm\"uller
space. Our main technical tool is a criterion, valid in any unique geodesic
metric space, that says that any geodesic ray with sufficiently many (in a
statistical sense) strongly contracting segments is sublinearly contracting.
