  We report observations of the linear polarization of a sample of 49 nearby
bright stars measured to sensitivities of between ~1 and 4 x 10^-6. The
majority of stars in the sample show measurable polarization, but most
polarizations are small with 75% of the stars having P < 2 x 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 x 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 x 10^-6 which could be due to
scattering in its dust disk, but is also consistent with interstellar
polarization in this direction. The highest polarization observed is that of BS
7405 (alpha Vul) with a polarization of 0.13%
