The complex sub and supercritical global dynamics of a parametrically excited microbeam is investigated with special consideration to chaotic motion. More specifically, for a microbeam subject to a time-dependent axial load involving a constant value together with a harmonic time-variant component, the bifurcation diagrams of Poincaré sections of the system near critical point are constructed when the amplitude of the longitudinal load variations is varied as the control parameter. In terms of modelling and simulations, the small-size-dependent potential energy of the system is constructed by means of the modified couple stress theory and constitutive relations. Continuous expressions for the kinetic energy and the energy dissipation mechanism are also constructed. A transformation to a high-dimensional reduced-order model is performed via use of an assumed-mode method as well as the Galerkin scheme. A direct time-integration method is employed to solve the reduced-order model. For different cases in the sub and supercritical regimes, but close to the critical mean axial force, the bifurcation diagrams of Poincaré sections are constructed as the amplitude of the axial load variations is chosen as the bifurcation parameter. The complex dynamical behaviour of the system is analysed more precisely through plotting time traces, fast Fourier transforms (FFTs), Poincaré sections and phase-plane diagrams.