Assuming that the underlying probability space is non-atomic,
we prove that products of random matrices (linear cocycles) with simple
Lyapunov spectrum form an $L^p$-dense set ($1 \leq p < \infty$) in the
space of all cocycles satisfying the integrability conditions of the
multiplicative ergodic theorem. However, the linear
cocycles with one-point spectrum are also $L^p$-dense.
Further, in any $L^\infty$-neighborhood of an orthogonal cocycle
there is a diagonalizable cocycle.
For products of independent identically
distributed random matrices (with distribution $\mu$), simplicity of
the Lyapunov spectrum holds on a set of $\mu$'s which is open and
dense in both the topology of total variation and the topology of weak
convergence, hence is generic in both topologies. For products of
matrices which form a Markov chain, the spectrum is simple on a set of
transition functions dense in the topology of weak convergence.