Abstract We prove that there exists a residual subset R (with respect to the C0 topology) of d-dimensional linear differential systems based in a μ-invariant flow and with transition matrix evolving in GL(d,ℝ) such that if A ∈ R, then, for μ-a.e. point, the Oseledets splitting along the orbit is dominated (uniform projective hyperbolicity) or else the Lyapunov spectrum is trivial. Moreover, in the conservative setting, we obtain the dichotomy: dominated splitting versus zero Lyapunov exponents.