Subject to a compressive membrane stress, an elastic film bonded on a substrate can become unstable, forming wrinkles, creases or delaminated buckles. Further increasing the compressive stress can induce advanced modes of instabilities including period-doubles, folds, localized ridges, delamination, and coexistent instabilities. While various instabilities in film-substrate systems under compression have been analyzed separately, a systematic and quantitative understanding of these instabilities is still elusive. Here we present a joint experimental and theoretical study to systematically explore the instabilities in elastic film-substrate systems under uniaxial compression. We use the Maxwell stability criterion to analyze the occurrence and evolution of instabilities analogous to phase transitions in thermodynamic systems. We show that the moduli of the film and the substrate, the film-substrate adhesion strength, the film thickness, and the prestretch in the substrate determine various modes of instabilities. Defects in the film-substrate system can facilitate it to overcome energy barriers during occurrence and evolution of instabilities. We provide a set of phase diagrams to predict both initial and advanced modes of instabilities in compressed film-substrate systems. The phase diagrams can be used to guide the design of film-substrate systems to achieve desired modes of instabilities.