The paper presents a new single-surface elasto-plastic model for unsaturated cemented soils, formulated within the critical state soil mechanics framework, which should be considered as an extension to unsaturated conditions of a recently proposed constitutive law for saturated structured soils. The model has been developed with the main purpose of inspecting the mechanical instabilities induced in natural soils by bond degradation resulting from the accumulation of plastic strains and/or the changes in pore saturation. At this scope, the constitutive equations are used to simulate typical geotechnical testing conditions, whose results are then analysed in light of the controllability theory. The results of triaxial tests on an ideal fully saturated cemented soil and on the corresponding unsaturated uncemented one are first discussed, aiming at detecting the evidence of potentially unstable conditions throughout the numerical simulations. This is followed by similar analyses considering the combined effects of both the above features. For each analysed case, a simple analytical stability criterion is proposed and validated against the numerical results, generalizing the results, and highlighting the crucial role of state variables and model parameters on the possible occurrence of failure conditions.