  Our knowledge of the magnetism in white dwarfs is based on an observational
dataset that is biased in favour of stars with very strong magnetic fields.
Most of the field measurements available in the literature have a relatively
low sensitivity, while current instruments allow us to detect magnetic fields
of white dwarfs with sub-kG precision. With the aim of obtaining a more
complete view of the incidence of magnetic fields in degenerate stars, we have
started a long-term campaign of high-precision spectropolarimetric observations
of white dwarfs. Here we report the results obtained so far with the
low-resolution FORS2 instrument of the ESO VLT and the medium-resolution ISIS
instrument of the WHT. We have considered a sample of 48 stars, of which five
are known magnetic or suspected magnetic stars, and obtained new longitudinal
magnetic field measurements with a mean uncertainty of about 0.6 kG. Overall,
in the course of our survey (the results of which have been partially published
in papers devoted to individual stars) we have discovered one new weak-field
magnetic white dwarf, confirmed the magnetic nature of another, found that a
suspected magnetic star is not magnetic, and suggested two new candidate
magnetic white dwarfs. Even combined with data previously obtained in the
literature, our sample is not sufficient yet to reach any final conclusions
about the actual incidence of very weak magnetic fields in white dwarfs, but we
have set the basis to achieve a homogeneous survey of an unbiased sample of
white dwarfs. As a by-product, our survey has also enabled us to carry out a
detailed characterisation of the ISIS and the FORS2 instruments for the
detection of extremely weak magnetic fields in white dwarfs, and in particular
to relate the S/N to measurement uncertainty for white dwarfs of different
spectral types. This study will help the optimisation of future observations.
