We have observed a sample of "ordinary" quasars, selected only on the grounds of their relative brightness in the visible, with the Turku Photopolarimeter on the 2.56m Nordic Optical Telescope, on La Palma, Canary Islands. The instrument permits us to obtain simultaneous UBVRI photometry and polarimetry of objects down to 16th magnitude, with the errors in the range <0.2-0.5%. Here we report results for an initial sample of sixteen objects, several of which have more than one observation. Our aim is to produce a sample of at least fifty objects which will allow us to examine the statistical properties of the polarization of ordinary quasars. Whilst our sample is still small for meaninful statistical analysis, various interesting trends are emerging. We find that, of 16 objects, 9 are detected at the 3 sigma level in at least two bands, all but one of which are radio quiet (ie: QSOs), whilst only one radio loud quasar (QSS) is detected. We also find that 7 of the quasars with detectable polarization are X-ray loud, whereas, of the non-detected quasars, 5 out of 7 are X-ray quiet and the same proportion are radio loud. Various objects show frequency dependence either of polarization degree, or position angle. Some evidence exists that nearly half of the sample may possibly have a polarization which varies by a factor of two in time scales of years.