Colloidal particles are often seen as big atoms that can be directly observed in real space. They are therefore becoming increasingly important as model systems to study processes of interest in condensed-matter physics such as melting, freezing and glass transitions. The solidification of colloidal suspensions has long been a puzzling phenomenon with many unexplained features. Here, we demonstrate and rationalize the existence of instability and metastability domains in cellular solidification of colloidal suspensions, by direct in situ high-resolution X-ray radiography and tomography observations. We explain such interface instabilities by a partial Brownian diffusion of the particles leading to constitutional supercooling situations. Processing under unstable conditions leads to localized and global kinetic instabilities of the solid/liquid interface, affecting the crystal morphology and particle redistribution behaviour.