We present size distributions of bubble and droplet populations generated by breaking waves using high-resolution direct numerical simulation of the two-phase Navier-Stokes equations with surface tension in three dimensions. We use the open-source code Basilisk with adaptive mesh refinement in order to achieve very high effective resolutions of up to 40963, resolving length scales ranging from the breaker wavelength down to sizes comparable to the bubble Hinze scale. The resulting solutions are feature-rich, generating statistical ensembles of bubble and droplet populations along with velocity fields for analysis of turbulence. In addition to the bubble and droplet statistics, we also discuss the role of transverse instabilities in the breakup of the large volumes of entrained air and the concomitant transition of turbulence from two to three dimensions in the breaker development. Finally, we discuss matters of numerical grid convergence with respect to the bubble and droplet size distributions at the smallest scales.