  This article investigates a few questions about orbits of local automorphisms
in manifolds endowed with rigid geometric structures. We give sufficient
conditions for local homogeneity in a broad class of such structures, namely
Cartan geometries, extending a classical result of Singer about locally
homogeneous Riemannian manifolds. We also revisit a strong result of Gromov
which describes the structure of the orbits of local automorphisms of manifolds
endowed with $A$-rigid structures, and give a statement of this result in the
setting of Cartan geometries.
