We report observations of the linear polarization of a sample of 49 nearby bright stars measured to sensitivities of between ∼1 and ∼ 4 × 10−6 . The majority of stars in the sample show measurable polarization, but most polarizations are small with 75 per cent of the stars having P < 2 × 10−5 . Correlations of the polarization with distance and position indicate that most of the polarization is of interstellar origin. Polarizations are small near the galactic pole and larger at low-galactic latitudes, and the polarization increases with distance. However, the interstellar polarization is very much less than would be expected based on polarization–distance relations for distant stars showing that the solar neighbourhood has little interstellar dust. BS 3982 (Regulus) has a polarization of ∼ 37 × 10−6 , which is most likely due to electron scattering in its rotationally flattened atmosphere. BS 7001 (Vega) has polarization at a level of ∼ 17 × 10−6 which could be due to scattering in its dust disc, but is also consistent with interstellar polarization in this direction. The highest polarization observed is that of BS 7405 (α Vul) with a polarization of 0.13 per cent.