  We investigate different geometrical properties of the inhomogeneous Poisson
point process $\Lambda_{\mu}$ associated to a positive, locally finite,
$\sigma$-finite measure $\mu$ on the unit disk. In particular, we characterize
the processes $\Lambda_{\mu}$ such that almost surely: 1) $\Lambda_{\mu}$ is a
Carleson-Newman sequence; 2) $\Lambda_{\mu}$ is the union of a given number M
of separated sequences. We use these results to discuss the measures $\mu$ such
that the associated process $\Lambda_{\mu}$ is almost surely an interpolating
sequence for the Hardy, Bloch or weighted Dirichlet spaces.
