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Aims. We have investigated a frequency-dependent shift in the absolute position of the optically thick apparent origin of parsec-scale jets ("core shift" effect) to probe physical conditions in ultra-compact relativistic outflows in active galactic nuclei. Methods. We used multi-frequency Very Long Baseline Array (VLBA) observations of 191 sources carried out in 12 epochs in 2006 within the Monitoring Of Jets in Active galactic nuclei with VLBA Experiments (MOJAVE) program. The observations were performed at 8.1, 8.4, 12.1, and 15.4 GHz. We implemented a method of determining the core shift vector based on (i) image registration by two-dimensional normalized cross-correlation and (ii) model-fitting the source brightness distribution to take into account a non-zero core component offset from the phase center. Results. The 15.4−8.1, 15.4−8.4, and 15.4−12.1 GHz core shift vectors are derived for 163 sources, and have median values of 128, 125, and 88 μas, respectively, compared to the typical measured errors of 50, 51, 35 μas. The effect occurs predominantly along the jet direction, with departures smaller than 45° from the median jet position angle in over 80% of the cases. Despite the moderate ratio of the observed frequencies (<2), core shifts significantly different from zero (>2σ) are detected for about 55% of the sources. These shifts are even better aligned with the jet direction, deviating from the latter by less than 30° in over 90% of the cases. There is an indication that the core shift decreases with increasing redshift. Magnetic fields in the jet at a distance of 1 parsec from the central black hole, calculated from the obtained core shifts, are found to be systematically stronger in quasars (median B1 ≈ 0.9 G) than those in BL Lacs (median B1 ≈ 0.4 G). We also constrained the absolute distance of the core from the apex of the jet at 15 GHz as well as the magnetic field strength in the 15 GHz core region. [ABSTRACT FROM AUTHOR]