Instabilities of convection in a mushy layer with a permeable interface underlying a liquid layer are studied in the time-dependent solidification system in which a binary melt cooled from below. The self-similar stability equations in the liquid and mushy layers are derived by propagation theory. The onset of mushy-layer-mode convection is examined considering the variation of permeability with porosity in the mushy layer. The numerical results show that the critical Darcy-Rayleigh number defined in terms of the mean permeability increases with increasing the concentration ratio and decreases with increasing the superheat. When the concentration ratio is small, a small convective cell appears in the vicinity of the liquid-mush interface. The influences of various non-uniform permeability models on the stability of compositional convection are discussed.