We analyse the circular polarisation data accumulated in the first 7 years of the POLAMI project introduced in an accompanying paper (Agudo et al.). In the 3mm wavelength band, we acquired more than 2600 observations, and all but one of our 37 sample sources were detected, most of them several times. For most sources, the observed distribution of the degree of circular polarisation is broader than that of unpolarised calibrators, indicating that weak (<0.5%) circular polarisation is present most of the time. Our detection rate and the maximum degree of polarisation found, 2.0%, are comparable to previous surveys, all made at much longer wavelengths. We argue that the process generating circular polarisation must not be strongly wavelength dependent, and we propose that the widespread presence of circular polarisation in our short wavelength sample dominated by blazars is mostly due to Faraday conversion of the linearly polarised synchrotron radiation in the helical magnetic field of the jet. Circular polarisation is variable, most notably on time scales comparable to or shorter than our median sampling interval <1 month. Longer time scales of about one year are occasionally detected, but severely limited by the weakness of the signal. At variance with some longer wavelength investigations we find that the sign of circular polarisation changes in most sources, while only 7 sources, including 3 already known, have a strong preference for one sign. The degrees of circular and linear polarisation do not show any systematic correlation. We do find however one particular event where the two polarisation degrees vary in synchronism during a time span of 0.9 years. The paper also describes a novel method for calibrating the sign of circular polarisation observations.
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