We present a Brownian dynamics simulation study of static and dynamic properties of quasi-two-dimensional dispersions of colloidal spheres interacting by long-range screened electrostatic and by dipolar magnetic forces, respectively. The calculated van Hove real-space dynamic correlation functions, mean squared displacements, and hydrodynamic functions are shown to obey a dynamic scaling behavior in terms of a characteristic relaxation time related to the geometrical mean particle distance. Hydrodynamic interactions introduce a second characteristic length scale, and they lead to a more restricted scaling behavior with an enhancement of self-diffusion. As a consequence of dynamic scaling, the dynamical criterion of Lowen [Phys. Rev. E 53, R29 (1996)] for the onset of colloidal freezing is shown to be equivalent to a two-dimensional freezing criterion related to the static structure factor.