The Mach evolution of the cylinder wake first and second bifurcations is investigated in the high-subsonic and supersonic regimes. A large database of two-dimensional (2D) and three-dimensional (3D) simulations was generated and validated by using two independent codes, named FaSTAR and OpenSBLI. The critical Reynolds number associated with the 2D periodic von Kármán vortex shedding is tracked up to Mach 0.9 by using FaSTAR and global stability analysis (GSA). The 3D span-periodic high-fidelity simulations carried out with OpenSBLI indicate that a spanwise modulation is superimposed to the 2D vortex shedding for sufficiently high Reynolds numbers at Mach 1.2. The simulations here presented demonstrate that the two bifurcations persist over a large range of flow speeds and, despite some differences in the unstable mode wavelengths, the mechanisms are essentially the same as those found in the incompressible regime. The generated database is also used to determine two analytical scaling functions for predicting separation length and critical Reynolds number around the first bifurcation. Both database and scaling functions can be used as a benchmark for the verification and validation of compressible CFD solvers over a large range of flow speeds.