We give sharp regularity results for the invariant subbundles of hyperbolic dynamical systems in terms of contraction and expansion rates and prove optimality in a strong sense: we construct open dense sets of codimension one systems where this regularity is not exceeded. Furthermore, we exhibit open dense sets of symplectic, geodesic, and codimension one systems where the analogous regularity results of [PSW] are optimal. As our main result we produce open sets of symplectic Anosov diffeomorphisms and flows with low transverse Hölder regularity of the invariant foliations almost everywhere. Prevalence of low regularity of conjugacies is a corollary. We also establish a new connection between the transverse regularity of foliations and their tangent subbundles.