In this proposal, researchers from Cascade Technologies and Stanford University outline a multi-year research plan to develop large-eddy simulation (LES) tools to predict and understand combustion instabilities in liquid-propellant rocket engines. Rocket instabilities are a notoriously complicated, multiscale problem involving nonlinear interactions between transcritical multiphase flows, turbulent mixing, combustion heat release, and acoustics. Each of these technical areas will be addressed to some extent during the course of the project. Central points of the Phase 1 plan include: adding real-fluid extensions to the open-source chemistry package Cantera, running CFD validation cases at rocket-relevant conditions, assessing the impact of low-dissipation numerical schemes on liquid sprays, and developing a unified multiphase formulation to span subcritical and supercritical conditions. These activities will set the stage for additional model developments and applied rocket simulations in Phase 2. Conclusions from the Phase 1 studies will help prioritize and plan the specific research areas to be addressed during Phase 2.