A Kelvin-Voigt based constitutive equation is implemented in Hamilton’s framework in order to derive the coupled in-plane/transverse equations governing the motion of a microplate with geometric imperfections, while considering geometric nonlinearities. The Kirchhoff plate theory and the modified couple stress-based theory (MCST) are utilized to obtain the strain and kinetic energies of the imperfect microsystem. Then, the Kelvin–Voigt energy dissipation scheme is employed to derive expressions for the work of the viscous components of the classical and non-classical stress tensors. Frequency-response diagrams are plotted to investigate the nonlinear resonant oscillations of the imperfect viscoelastic microsystem in the presence of geometric imperfections. Numerical simulations revealed that the concurrent presence of geometric imperfections and the nonlinear amplitude-dependent damping mechanism alters the bifurcational behaviour of the viscoelastic microsystem substantially. It is shown that at oscillations of large amplitude, the nonlinear damping contributions become significant.