We present a mathematical model of the directional freezing of colloidal suspensions. Key ingredients of the model are the disjoining forces between the colloidal particles and the solidified suspending fluid, flow of the suspending fluid towards the solidification front through an accumulating layer of particles, and flow through microscopic films of unfrozen liquid separating particles from the freezing front. Our model predicts three dif- ferent modes of solidification leading to different microstructures: dendritic formations; laddered structures of ice spears and lenses; a frozen fringe, from which transverse ice lenses can form. It explains why different researchers have reported the existence of ice lensing with and without the pre-existence of frozen fringes. Our quantitative predictions are encapsulated within a universal, dimensionless phase diagram showing which mi- crostructure is to be expected under which operating conditions, and we show that these predictions are consistent with previous experimental studies as well as new experiments that we present here.