Complete regularity of linear cocycles and the Baire category of the set of Lyapunov-Perron regular points

Given a continuous linear cocycle $\mathcal{A}$ over a homeomorphism $f$ of a compact metric space $X$, we investigate its set $\mathcal{R}$ of Lyapunov-Perron regular points, that is, the collection of trajectories of $f$ that obey the conclusions of the Multiplicative Ergodic Theorem. We obtain results roughly saying that the set $\mathcal{R}$ is of first Baire category (i.e., meager) in $X$, unless some rigid structure is present. In some settings, this rigid structure forces the Lyapunov exponents to be defined everywhere and to be independent of the point; that is what we call complete regularity.