The electron density profiles at monovacancies of simple metals are calculated by the self-consistent Kohn-Sham method and by a number of statistical methods. The metal is described by a uniform positive background charge together with an interacting electron gas, and the vacancy is approximated as a spherical hole in the background. The Kohn-Sham electron density inside the vacancy is found to be in average 15 of the density in the bulk material. Of the various statistical methods, the simple Thomas-Fermi approximation is found to describe best the average electron density over the whole metallic density range when compared to the Kohn-Sham results. The energies of vacancy formation are calculated by using the Kohn-Sham electron densities together with three lattice models, and reasonable numerical success is achieved for alkali metals. In the case of polyvalent metals the results are not satisfactory even if the uniform background were replaced by point ions or if the electron-ion interactions were described by Ashcroft empty-core pseudopotentials. The lifetimes of a positron trapped at the vacancies of several metals are calculated by using both the Kohn-Sham and the Thomas-Fermi electron densities. The results for most metals are in agreement with experimental values. The angular-correlation curve of the positron in aluminum vacancy is calculated directly from the Kohn-Sham one-electron wave functions. The result agrees with the curve calculated from the so-called mixed-density approximation and also with the experimental result.