A dynamic linear instability analysis of steady-state binary melt solidification with a mushy region has been carried out. Such an instability differs from a conventional morphological one of a planar solid-liquid interface and is connected with the perturbations in the steady-state solidification velocity. Solidification with a narrow mushy region has been revealed to be absolutely unstable with a monotonous instability of a "hard" type. An increase of the mushy region width leads to an instability changeover from the "hard" type to the oscillatory "soft" one. Both the critical changeover width and the neutral stability curves have been determined as the functions of relevant physical and operating parameters. The steady-state solidification regime with a broad mushy region is absolutely stable. Thus, the mushy region width has been shown to represent a stabilizing factor.