  Random walks on spaces with hyperbolic properties tend to sublinearly track
geodesic rays which point in certain hyperbolic-like directions.
Qing-Rafi-Tiozzo recently introduced the sublinearly Morse boundary and proved
that this boundary is a quasi-isometry invariant which captures a notion of
generic direction in a broad context.
  In this article, we develop the geometric foundations of sublinear Morseness
in the mapping class group and Teichm\"uller space. We prove that their
sublinearly Morse boundaries are visibility spaces and admit continuous
equivariant injections into the boundary of the curve graph. Moreover, we
completely characterize sublinear Morseness in terms of the hierarchical
structures of these spaces.
  Our techniques include developing tools for modeling the hulls of median rays
in hierarchically hyperbolic spaces via CAT(0) cube complexes. Part of this
analysis involves establishing direct connections between the geometry of the
curve graph and the combinatorics of hyperplanes in the approximating cube
complexes.
