The International Muon Collider Collaboration (IMCC) is investigating the key challenges of a 10 TeV centre-of-mass muon collider ring, along with its injector complex and an intermediate 3 TeV collider stage. Muon and anti-muon bunches are produced via a proton driver complex and then undergo a process called 6D cooling. The bunches are then accelerated by a series of Linacs, Recirculating Linacs (RLA) and Rapid Cycling Synchrotrons (RCS) before entering the collider ring. Collective effects are a concern due to the high charge of the muon bunches. The RCS require a significant number of RF cavities to rapidly accelerate the beams and keep a 90 % survival rate in each ring. The effect of the cavity high-order modes (HOMs) was evaluated using start-to-end simulations that included collective effects. The collider would be an isochronous ring to preserve a short bunch length. The study also examined the impact of this operation on transverse coherent stability, and potential methods for mitigating instabilities.