  It follows from Oseledec Multiplicative Ergodic Theorem that the
Lyapunov-irregular set of points for which the Oseledec averages of a given
continuous cocycle diverge has zero measure with respect to any invariant
probability measure. In strong contrast, for any dynamical system
$f:X\rightarrow X$ with exponential specification property and a
H$\ddot{\text{o}}$lder continuous matrix cocycle $A:X\rightarrow G
(m,\mathbb{R})$, we show here that if there exist ergodic measures with
different Lyapunov spectrum, then the Lyapunov-irregular set of $A$ is residual
(i.e., containing a dense $G_\delta$ set).
