Swirl injectors has been broadly used in contemporary liquid-propellant rocket engines to achieve efficient mixing and combustion, but the understanding of injector flow physics and dynamics at supercritical conditions is still very limited. This paper intends to improve such understanding by carrying out a systematic three-dimensional numerical study of simplex swirl injector. The numerical scheme is based on full-conservation laws and accommodates real-fluid thermodynamics and transport theory over entire range of fluid states. The complex three-dimensional flow structures and dynamics are visualized and explored for the first time using the spectral analysis and proper orthogonal decomposition technique at supercritical conditions. A transition region from the compressed-liquid to light-gas state is observed at supercritical pressures. Various underlying mechanisms dictating the flow evolution, including Kelvin-Helmholtz instability, centrifugal instability, center-recirculating flow, helical instability, and their interactions are studied. The longitudinal hydrodynamic instability is found to play a dominant role in the oscillatory flowfield. The third-mode transverse instability is amplified and resonant with the acoustic wave at the characteristic frequency of 4.8 kHz. The converging section reflects the waves back into the vortex chamber and only allows part of waves with long wavelengths to transmit to the discharge nozzle. A parametric study is made to examine the pressure and temperature effects on the injector design attributes, such as film thickness and spray cone angle. The results are also compared to predictions from classical hydrodynamics theories to acquire direct insight into the flow physics involved. The current study will provide the benchmark for the future research on the mixing and combustion of swirling injection flows.