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Transformation of the whispering-gallery (WG)-modes in imperfect two-dimensional square and triangular microcavities with various degrees of deformation is studied by means of the Muller boundary integral equation technique. Continuous transformation of a circular microcavity, which supports double-degenerate WG-modes, toward a square or a triangle removes the degeneracy of certain WG-modes. The spectrum of the modes that split depends on the symmetry of the emerging microcavity: WG2m,n-modes in case of a C4v symmetry (square) or WG3m,n-modes in the case of C3v symmetry (triangle). In both cases, the modes with the highest Q-factors are nondegenerate modes with antisymmetrical field patterns. We estimate mode frequencies, quality factors and field distributions of practically achievable rather than "ideal" square and triangular microcavities, and compare the effect of various types of fabrication imperfections (corner sharpness, sidewall curvature and surface roughness) on their characteristics. Accurate study of the modal spectra enables us to confirm and explain previous observations, such as: 1) co-existence of the WG-like and volume modes in square microcavities; 2) the separation of the high-Q WG-like modes being twice that determined by the cavity length; and 3) much lower Q-factors of realistic concave-wall triangular microcavities than those of their ideal counterparts.