Instabilities that arise in layered systems have been a riveting course of study for the past few decades, having found utility in various fields, while also being frequently observed in biological systems. However, the large deformation bifurcation analysis of compressible domains remains vastly understudied compared to the incompressible case. In this work, we present a unifying approach for the instability analysis of multilayer compressible elastic domains under plane deformations, also extending the approach to include the general interface model and to capture growth-induced instabilities. First, a linear elastic, displacement-based approach to capture bilayer wrinkling is taken, outlining the basics of such an approach. Then, the large-deformation incremental analysis for a rectangular, compressible, hyperelastic domain under plane deformations is developed, which serves as a generic framework for other geometries. This framework is applied to beam, half-space, bilayer and trilayer structures. Next, the framework is extended to account for the general interface model, looking into coated half-spaces, coated beams, and bilayers with interfaces. Finally, the framework is derived to account for both compression and growth. Obtained analytical results for the onset of wrinkling are compared with computational simulations using the finite element method (FEM) enhanced with eigenvalue analysis, cultivating excellent agreements between analytical and numerical results all across the material and geometrical parameter spectrum, and portraying clearly the significant effect of compressibility on bifurcation behavior. The analytical framework presented here provides grounds for further works on other modes of instabilities and more complex geometries.