This paper presents a novel methodology for detecting and analyzing oscillation modes resulting from interactions between synchronous machines (SMs) and inverterbased resources (IBRs). The methodology utilizes a conceptual benchmark specifically designed to identify various oscillation modes, particularly those involving SMs and IBRs in a minimal order model for any power system. These phenomena can be studied by adapting the benchmark to represent any power system- in this paper a French island power system- where the IBR is controlled in grid-forming mode. In this system, oscillations among its generation sources raise concerns for the grid operator, necessitating comprehensive studies to understand potential stability and safety risks. The benchmark is linearised around an operating point and the type of each detected mode is identified through calculation of participation factors. The impact of key parameters on the damping of the modes is studied through sensitivity analyses. Nonlinear time-domain simulations, based on electromagnetic transient (EMT) modeling using EMTP-RV software are employed to validate the linearised model of the benchmark. Finally, a power oscillation damping (POD) controller installed on the inverter is proposed to mitigate critical oscillation modes.