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 0.6kG. 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 signal-to-noise ratio to measurement uncertainty for white dwarfs of different spectral types. This study will help the optimisation of future observations.