In recent years, single-photon sources (SPSs) based on the emission of a single semiconductor quantum dot (QD) have been actively developed. While the purity and indistinguishability of single photons are already close to ideal values, the high brightness of SPSs remains a challenge. The widely used resonant excitation with cross-polarization filtering usually leads to at least a two-fold reduction in the single-photon counts rate, since single-photon emission is usually unpolarized, or its polarization state is close to that of the exciting laser. One of the solutions is the use of polarization-selective microcavities, which allows one to redirect most of the QD emission to a specific polarization determined by the optical mode of the microcavity. In the present work, elliptical micropillars with distributed Bragg reflectors are investigated theoretically and experimentally as a promising design of such polarization-selective microcavities. The impact of ellipticity, ellipse area and verticality of the side walls on the splitting of the optical fundamental mode is investigated. The study of the near-field pattern allows us to detect the presence of higher-order optical modes, which are classified theoretically. The possibility of obtaining strongly polarized single-photon QD radiation associated with the short-wavelength fundamental cavity mode is shown.