We discuss the relation between three recent approaches of describing the dynamics and the spatial distribution of particles suspended in turbulent flows: phase-space singularities in the inertial particle dynamics (caustics), real-space singularities of the deformation tensor, and random uncorrelated motion. We discuss how the phase- and real-space singularities are related. Their formation is well understood in terms of a local theory. We discuss implications for random uncorrelated motion. Our results are supported by results of direct numerical simulations of inertial particles in model flows. ; Comment: 20 pages, 9 figures, as published