A tiny Lorentz symmetry breaking can be mediated in Electrodynamics by means of the Chern-Simons (CS) interaction polarized along a constant CS vector. Its presence makes the vacuum optically active that has been recently estimated from astrophysical data. We examine two possibilities for the CS vector to be time-like or space-like, under the assumption that it originates from v.e.v. of some pseudoscalar matter. It is shown that: a) a time-like CS vector makes the vacuum unstable under pairs creation of tachyonic photon modes with the finite vacuum decay rate, i.e. it is unlikely realized at macroscopic time scales; b) on the contrary, the space-like CS vector does not yield any tachyonic modes and, moreover, if its dynamical counterpart is substantially described by a scale invariant interaction, then the QED radiation effects induce the dynamical breaking of Lorentz symmetry, i.e. the occurrence of space-like CS vector appears to be rather natural.