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

The geometry of genericity in mapping class groups and Teichm\"uller spaces via CAT(0) cube complexes