The stability characteristics of a class of unstructured models of continuous bioreactors are analyzed using elementary concepts of singularity theory and continuation techniques. The class consists of models for which the non-biomass product formation rate is linearly proportional to the utilization rate of limiting substrate. The kinetics expressions of cell growth and product synthesis are allowed to assume general forms of substrate and product. Global analytical conditions are derived that allow the construction of a practical picture in the multidimensional parameter space delineating the different static behavior these models can predict, including unique steady states, coexistence of non-trivial steady states with wash-out conditions, and multistability resulting from hysteresis. These general results are applied to specific examples of bioprocesses and allow the study of the effect of kinetic and operating parameters on the stability characteristics of these models.