An effective high-accuracy method for finding surface states or surface resonances of a given crystal potential, based on locating singularities of the scattering matrix of the semi-infinite crystal, is described and illustrated by detailed calculations. The method is applied to Al(001) using a potential that is known to give reasonable results for bulk and surface properties; the effects of various transitional potentials between bulk and vacuum are also studied. A large number and a great variety of surface states are found, which can be classified by the types and numbers of overlapping energy gaps of the bulk band structure in which they occur. Seven types are found between the muffin-tin zero and the vacuum level, and surface states are present in almost all the energy gaps in that energy range. For the most widely distributed types in the two-dimensional Brillouin zone, densities of surface states in energy are found that include logarithmic singularities. A convenient method is described for calculating such densities of states for states whose energies are given on an arbitrary mesh of points in two dimensions. © 1979 The American Physical Society.