The multi-timescale interaction between wind turbines and the power grid in direct-drive wind power systems is prone to inducing sub-/super-synchronous oscillations, necessitating the exploration of effective suppression methods. Flywheel energy storage, with its advantages in fast dynamic power response and high charging/discharging frequency per unit time, can enhance system damping characteristics and holds potential for suppressing sub-/super-synchronous oscillations. This paper proposes an adaptive damping suppression method against sub-/super-synchronous oscillations based on flywheel energy storage. A sub-/super-synchronous oscillation adaptive damper (SSOAD) is designed and integrated into the grid-side converter of the flywheel energy storage system, leveraging its capability to rapidly absorb and release energy to achieve sub-/super-synchronous oscillation suppression. First, an overall architecture of SSOAD integrating “measurement-identification-control” functions is proposed. On this basis, a sub-/super-synchronous oscillation detection and identification method, along with an adaptive multi-channel damper, is designed to identify the number of oscillation modes and their respective frequencies. Then, a control strategy for the flywheel energy storage system is designed, including an oscillation suppression method based on current compensation and a bidirectional charging/discharging conversion control method. Finally, a simulation model of a direct-drive wind power system is built to validate the feasibility of the proposed SSOAD. The results demonstrate that the designed SSOAD not only enables real-time identification of multiple oscillation modes, but also ensures effective suppression of sub-/super-synchronous oscillations under varying operation conditions.