Aims: We completed observations with the Effelsberg 100-m radio telescope to measure the polarised emission from a complete sample of compact steep-spectrum sources and improve our understanding of the physical conditions inside and around regions of radio emission embedded in dense interstellar environments. Methods: We observed the sources at four different frequencies, namely 2.64 GHz, 4.85 GHz, 8.35 GHz, and 10.45 GHz, making use of the polarimeters available at the Effelsberg telescope. We complemented these measurements with polarisation parameters at 1.4 GHz derived from the NRAO VLA Sky Survey. Previous single dish measurements were taken from the catalogue of Tabara and Inoue. Results: The depolarisation index DP was computed for four pairs of frequencies. A drop in the fractional polarisation appeared in the radio emission when observing at frequencies below ~2 GHz. Rotation measures were derived for about 25% of the sources in the sample. The values, in the source rest frame, range from about -20 rad m-2 found for 3C 138 to 3900 rad m-2 in 3C 119. In all cases, the λ2 law is closely followed. Conclusions: The presence of a foreground screen as predicted by the Tribble model or with “partial coverage” as defined by ourselves can explain the polarimetric behaviour of the CSS sources detected in polarisation by the present observations. Indication of repolarisation at lower frequencies was found for some sources. A case of possible variability in the fractional polarisation is also suggested. The most unexpected result was found for the distribution of the fractional polarisations versus the linear sizes of the sources. Our results appear to disagree with the findings of Cotton and collaborators and Fanti and collaborators for the B3-VLA sample of CSS sources, the so-called “Cotton effect”, i.e., a strong drop in polarised intensity for the most compact sources below a given frequency. This apparent contradiction may, however, be caused by the large contamination of the sample by quasars with respect to the B3-VLA. Tables [see full textsee full textsee full text]-[see full textsee full textsee full text] and Appendices are only available in electronic form at http://www.aanda.org