We present new optical broad-band (UBVRI) aperture polarimetric observations of 53 post-asymptotic giant branch (AGB) stars selected to exhibit a large near-infrared excess. 24 out of the 53 stars (45% of our sample) are presented for the first time. A statistical analysis shows four distinctive groups of polarized post-AGB stars: unpolarized or very lowly polarized (degree of polarization or DoP<1%), lowly polarized (1%<DoP<4%), moderately polarized (4%<DoP<8%) and highly polarized (DoP>8%). 23 out of the 53 (66%) belong to the first group, 10 (19%) to the second, five ( %) to the third and only three (6%) to the last group. Approximately 34%of our sample was found to be unpolarized objects, which is close to the percentage of round planetary nebulae. On average, the low and moderate groups show a wavelength-dependent polarization that increases towards shorter wavelengths, implying an intrinsic origin of the polarization, which signifies a Rayleigh-like scattering spectrum typical for non-symmetrical envelopes composed principally of small dust grains. The moderately polarized stars exhibit higher K-W3 and W1-W3 colour indices compared with the group of lowly polarized stars, suggesting a possible relation between DoP and mass-loss rate. Moreover, they are found to be systematically colder (redder in B-V), which may be associated with the condensation process close to these stars that results in a higher degree of polarization. We also provide evidence that multiple scattering in optically thin polar outflows is the mechanism that gives high DoP in post-AGB stars with bipolar or multi-polar envelopes.