Summary: A perturbation expansion is offered for the micromechanical prediction of the bifurcation buckling of soft viscoelastic composites with imperfections (e.g. wavy fibers). The composites of periodic microstructure are subjected to compressive loading and are undergoing large deformations. The perturbation expansion applied on the imperfect composites results in a zero and first order problems of perfect composites. In the former problem, loading exists and interfacial and periodicity conditions are imposed. In the latter one, however, loading is absent, the interfacial conditions possess complicated terms that have been already established by the zero order problem, and Bloch-Floquet boundary conditions are imposed. Both problems are solved by the high-fidelity generalized method of cells (HFGMC) micromechanical analysis. The ideal critical bifurcation stress can be readily predicted from the asymptotic values of the form of waviness growth with applied loading. This form enables also the estimation of the actual critical stress. The occurrence of the corresponding critical deformation and time is obtained by generating the stress-deformation response of the composite. The offered approach is illustrated for the prediction of bifurcation buckling of viscoelastic bi-layered and polymer matrix composites as well as porous materials. Finally, bifurcation buckling stresses of unidirectional composites in which the matrix is represented by the quasi-linear viscoelasticity theory are predicted. This quasi-linear viscoelasticity model exhibits constant damping which is observed by the actual viscoelastic behavior of biological materials.