Context. In November 2021, Solar Orbiter started its nominal mission phase. The remote-sensing instruments on board the spacecraft acquired scientific data during three observing windows surrounding the perihelion of the first orbit of this phase. Aims. The aim of the analysis is the detection of magnetohydrodynamic (MHD) wave modes in an active region by exploiting the capabilities of spectropolarimetric measurements. Methods. The High Resolution Telescope (HRT) of the Polarimetric and Helioseismic Imager (SO/PHI) on board the Solar Orbiter acquired a high-cadence data set of an active region. This is studied in the paper. B-ω and phase-difference analyses are applied on line-of-sight velocity and circular polarization maps and other averaged quantities. Results. We find that several MHD modes at different frequencies are excited in all analysed structures. The leading sunspot shows a linear dependence of the phase lag on the angle between the magnetic field and the line of sight of the observer in its penumbra. The magnetic pore exhibits global resonances at several frequencies, which are also excited by different wave modes. Conclusions. The SO/PHI measurements clearly confirm the presence of magnetic and velocity oscillations that are compatible with one or more MHD wave modes in pores and a sunspot. Improvements in modelling are still necessary to interpret the relation between the fluctuations of different diagnostics.