  Selection of extragalactic point sources, e.g. QSOs, is often hampered by
significant selection effects causing existing samples to have rather complex
selection functions. We explore whether a purely astrometric selection of
extragalactic point sources, e.g. QSOs, is feasible with the ongoing Gaia
mission. Such a selection would be interesting as it would be unbiased in terms
of colours of the targets and hence would allow selection also with colours in
the stellar sequence. We have analyzed a total of 18 representative regions of
the sky by using GUMS, the simulator prepared for ESAs Gaia mission, both in
the range of $12\le G \le 20$ mag and $12\le G \le 18$ mag. For each region we
determine the density of apparently stationary stellar sources, i.e. sources
for which Gaia cannot measure a significant proper motion. The density is
contrasted with the density of extragalactic point sources, e.g. QSOs, in order
to establish in which celestial directions a pure astrometric selection is
feasible. When targeting regions at galactic latitude $|b| \ge 30^\mathrm{o}$
the ratio of QSOs to apparently stationary stars is above 50\% and when
observing towards the poles the fraction of QSOs goes up to about $\sim80$\%.
We show that the proper motions from the proposed Gaia successor mission in
about 20 years would dramatically improve these results at all latitudes.
Detection of QSOs solely from zero proper motion, unbiased by any assumptions
on spectra, might lead to the discovery of new types of QSOs or new classes of
extragalactic point sources.
