The influence of the inhomogeneity of material properties on the process of three-dimensional stability loss of a hollow cylinder stretched by axial force and loaded by uniform pressure on the outer or inner side surface is investigated. We used two standard models describing the compressible nonlinearly elastic material's mechanical properties, namely the three-constant Blatz and Ko model, as well as the five-constant Mournaghan model. Usage of the semi-inverse method allows the reduction of a three-dimensional cylinder equilibrium problem to the study of a non-linear boundary-value problem for an ordinary second-order differential equation. Stability analysis was carried out based on the linearization of the equilibrium equations in the vicinity of the constructed solution. The value of a de-formation characteristic for which there were nontrivial solutions of a homogeneous boundary-value problem for the equations of neutral equilibrium obtained in the linearization process was identified with the critical value of the loading parameter, i.e., the value at which the system loses stability. The coefficients of the cylinder's stretching or radial expansion and the dimensionless characteristic of the applied pressure served as such parameters. On the plane of the loading parameters, stability regions are determined. The influence of heterogeneity on the size and shape of these regions is analyzed.