Regularity of abelian linear actions

We study the regularity of orbits for the natural action of a Lie subgroup G of GL(V), where V is a finite dimensional real vector space. When G is connected, abelian, and satisfies a certain rationality condition, we show that there are two possibilities: either there is a G-invariant Zariski open set Omega in which every orbit is regular, or there is a G-invariant conull S-delta set in which every orbit is not regular. Moreover, under the rationality condition, an explicit characterization of almost everywhere regularity is proved.