A three-dimensional buckling delamination problem for the sandwich rectangular plate made from elastic and viscoelastic material is studied. It is supposed that the plate contains interface rectangular cracks (Case 1) and interface rectangular edge-cracks (Case 2) and edge-surfaces of these cracks have initial infinitesimal imperfections. The evolution of these initial imperfections with an external compressive loading acting along the cracks (for a case where the materials of layers of the plate are elastic) or with duration of time (for a case where the materials of layers of the plate are viscoelastic) is investigated within the framework of the piecewise homogeneous body model with the use of three-dimensional geometrically nonlinear field equations of the theory of the viscoelastic bodies. For the determination of the values of the critical force or critical time as well as the buckling delamination mode, the initial imperfection criterion is used. The corresponding boundary-value problems are solved by employing boundary form perturbation techniques, Laplace transform and FEM. The influence of the materials or geometrical parameters of the plate on the critical values is discussed. In particular, it is established that for the considered change range of the problem parameters the buckling form depends not only on the infinitesimal initial imperfection mode of the crack edges, but also on the parameters which characterize the geometry and location of these cracks.