  We develop a paradifferential approach for studying non-smooth hyperbolic
dynamics and related non-linear PDE from a microlocal point of view. As an
application, we describe the microlocal regularity, i.e the $H^s$ wave-front
set for all $s$, of the unstable bundle $E_u$ for an Anosov flow. We also
recover rigidity results of Hurder-Katok and Hasselblatt in the Sobolev class
rather than H\"older: there is $s_0>0$ such that if $E_u$ has $H^s$ regularity
for $s>s_0$ then it is smooth (with $s_0=2$ for volume preserving
$3$-dimensional Anosov flows). In the appendix by Guedes Bonthonneau, it is
also shown that it can be applied to deal with non-smooth flows and potentials.
This work could serve as a toolbox for other applications.
