[ABRIDGED] We have carried out a systematic search for close supermassive black hole binaries among z < 0.7 SDSS quasars Such binaries are predicted by models of supermassive black hole and host galaxy co-evolution, therefore their census and population properties constitute an important test of these models. We used an automatic technique based on spectroscopic principal component analysis to search for broad H-beta lines that are displaced from the rest-frame of the quasar by more than 1,000 km/s This method can also yield candidates for rapidly recoiling black holes. Our search yielded 88 candidates, several of which were previously identified and discussed in the literature. The widths of the broad H-beta lines are typical among quasars but the shifts are extreme. We found a correlation between the peak offset and skewness of the broad H-beta profiles, which suggests that the profiles we have selected share a common physical explanation. The general properties of the narrow emission lines are typical of quasars. We carried out followup spectroscopic observations of 68 objects to search for changes in the peak velocities of the H-beta lines (the time interval in the observers frame between the original and new observations is 1-10 yr). We measured significant changes in 14 objects, with resulting accelerations between -120 and +120 km/s/yr. We emphasize that interpretation of the offset broad emission lines as signatures of supermassive binaries is subject to many significant caveats. Many more followup observations over a long temporal baseline are needed to characterize the variability pattern of the broad lines and test that this pattern is indeed consistent with orbital motion. The possibility that some of the objects in this sample are rapidly recoiling black holes remains open as the available data do not provide strong constraints for this scenario.
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