The most famous case of polyamorphism is amorphous ice. Pressurizing conventional hexagonal ice crystals to about 1.6 GPa at liquid nitrogen temperature (77 K) converts them to the high-density amorphous ice. Upon releasing the pressure, this phase is stable and has density of 1.17 g/cm at 77 K and 1 bar. Consequent warming to 127 K at ambient pressure transforms this phase to a low-density amorphous ice (0.94 g/cm at 1 bar). Yet, if the high-density amorphous ice is warmed up to 165 K not at low pressures but keeping the 1.6 GPa compression, and then cooled back to 77 K, then another amorphous ice is produced, which has even higher density of 1.25 g/cm at 1 bar. All those amorphous forms have very different vibrational lattice spectra and intermolecular distances. 
A similar abrupt liquid-amorphous phase transition is predicted in liquid silicon when cooled under high pressures. This observation is based on first principles molecular dynamics computer simulations, and might be expected intuitively since tetrahedral amorphous carbon, silicon, and germanium are known to be structurally analogous to water.