Context. The identification of the carriers of diffuse interstellar bands (DIBs) remains one of the long-standing mysteries in astronomy. The detection of a polarisation signal in a DIB profile can be used to distinguish between a dust or gas-phase carrier. The polarisation profile can give additional information on the grain or molecular properties of the absorber. Aims. To measure the polarisation efficiency of the carriers of the diffuse interstellar bands. Methods. In order to detect and measure the linear and circular polarisation of the DIBs we observed reddened lines of sight showing continuum polarisation. For this study we selected two stars HD 197770 and HD 194279. We used high-resolution (R ∼ 64 000) spectropolarimetry in the wavelength range from 3700 to 10 480 Å with the ESPaDOnS échelle spectrograph mounted at the CFHT. Results. High S/N and high resolution Stokes V (circular), Q and U (linear) spectra were obtained. We constrained upper limits by a factor of 10 for previously observed DIBs. Furthermore, we analysed ∼30 additional DIBs for which no spectropolarimetry data has been obtained before. This included the 9577 Å DIB and the 8621 Å DIB. The former is attributed to the C + 60 fullerene, which could become aligned in a magnetic field. The latter shows a tight correlation with the amount of dust in the line-of-sight and therefore most likely may show a polarisation signal related the aligned grains. Conclusions. The lack of polarisation in 45 DIB profiles suggests that none of the absorption lines is induced by a grain-type carrier. The strict upper limits, less than ∼0.01%, derived for the observed lines-of-sight imply that if DIBs are due to gas-phase molecules these carriers have polarisation efficiencies which are at least 6 times, and up to 300 times, smaller than those predicted for grainrelated carriers.