We study the band dispersion of graphene with randomly distributed structural defects using two complementary methods, exact diagonalization of the tight-binding Hamiltonian and self-consistent T-matrix approximation. We identify two distinct types of impurities--weak and resonant--resulting in qualitatively different spectra in the vicinity of the Dirac point. Weak impurities, such as paired vacancies or Stone-Wales defects, provide a line broadening that logarithmically increases with energy. Resonant impurities, such as vacancies or 585 defects, lead to stretching of the spectrum near the Dirac point with a finite density of localized states. This type of spectrum has been observed in epitaxial graphene by photoemission spectroscopy and discussed extensively in the literature. We find a good agreement between the results of the two theoretical methods and also with the experimentally measured spectra.