We conducted a spectropolarimetic survey of 58 high proper-motion white dwarfs which achieved uncertainties of >rsim 2 kG in the Hα line and >rsim 5 kG in the upper Balmer line series. The survey aimed at detecting low magnetic fields (≲ 100 kG) and helped identify the new magnetic white dwarfs NLTT 2219, with a longitudinal field Bl = −97 kG, and NLTT 10480 (Bl = −212 kG). Furthermore, we report the possible identification of a very low-field white dwarf with Bl = −4.6 kG. The observations show that ≈5 per cent of white dwarfs harbour low fields (∼10 to ∼102 kG) and that increased survey sensitivity may help uncover several new magnetic white dwarfs with fields below ∼1 kG. A series of observations of the high-field white dwarf NLTT 12758 revealed changes in polarity occurring within an hour possibly associated with an inclined, fast rotating dipole. Also, the relative strength of the π and σ components in NLTT 12758 possibly revealed the effect of a field concentration (‘spot’), or, most likely, the presence of a non-magnetic white dwarf companion. Similar observations of NLTT 13015 also showed possible polarity variations, but without a clear indication of the time-scale. The survey data also proved useful in constraining the chemical composition, age and kinematics of a sample of cool white dwarfs as well as in constraining the incidence of double degenerates.