Non-linear bifurcation theory and numerical continuation of bifurcations are important methods to predict the oscillation evolution process of a hydro-turbine governing system. The system's oscillation characteristic is directly related to three factors, namely the generator damping, excitation gain and proportion-integration-differentiation controller. Accordingly, three typical bifurcation and continuation scenarios related to these factors are studied, based on a non-linear dynamical model of the governing system in which the excitation system and the power system stabiliser are included. Some important non-linear dynamic phenomena, such as the equilibrium curves stability, bifurcation points location and limit cycle direction, are exhaustively depicted. Moreover, the dynamic behaviour of the system near bifurcation points is also illustrated through both time-domain simulation results and phase trajectory diagrams. The results show that bifurcations of more and more complicated types are found starting from simple objects like equilibria, which is an important route to study the system's dynamic behaviour. An interesting aspect is that the hydro-turbine governing system exhibits multistability, i.e. for some parameter value sets, there is a non-connected set of stable equilibria. Finally, these results provide a predicted reference for the parameter setting to ensure the stability and safety of the hydro-turbine governing system.