The empirical distribution of projected rotational velocities (vsini) in massive O-type stars is characterized by a dominant slow velocity component and a tail of fast rotators. Binary interaction has been proposed to play a dominant role in the formation of this tail. We perform a complete and homogeneous search for empirical signatures of binarity in a sample of 54 fast-rotating stars with the aim of evaluating this hypothesis. This working sample has been extracted from a larger sample of 415 Galactic O-type stars which covers the full range of vsini values. We used new and archival multi-epoch spectra in order to detect spectroscopic binary systems. We complement this information with Gaia proper motions and TESS photometric data to aid in the identification of runaway stars and eclipsing binaries, respectively. We also benefitted from additional published information to provide a more complete overview of the empirical properties of our working sample of fast-rotating O-type stars. The identified fraction of single-lined spectroscopic binary (SB1) systems and apparently single stars among the fast-rotating sample is ~18% and ~70%, respectively. The remaining 12% correspond to four secure double-line spectroscopic binaries (SB2) with at least one of the components having a vsini>200km/s (~8%), along with a small sample of 2 stars (~4%) for which the SB2 classification is doubtful: these could actually be single stars with a remarkable line-profile variability. When comparing these percentages with those corresponding to the slow-rotating sample, we find that our sample of fast rotators is characterised by a slightly larger percentage of SB1 systems (~18% vs. ~13%) and a considerably smaller fraction of clearly detected SB2 systems (8% vs. 33%). Overall, there seems to be a clear deficit of spectroscopic binaries (SB1+SB2) among fast-rotating O-type stars (~26% vs. ~46%). On the contrary, the fraction of runaway stars is significantly higher in the fast-rotating domain (~33-50%) than among those stars with vsini<200km/s. Lastly, almost 65% of the apparently single fast-rotating stars are runaways. As a by-product, we discovered a new over-contact SB2 system (HD 165921) and two fast-rotating SB1 systems (HD 46485 and HD 152200) Also, we propose HD 94024 and HD 12323 (both SB1 systems with a vsini<200km/s) as candidates for hosting a quiescent stellar-mass black hole. Our empirical results seem to be in good agreement with the assumption that the tail of fast-rotating O-type stars (with vsini>200km/s) is mostly populated by post-interaction binary products. In particular, we find that the final statistics of identified spectroscopic binaries and apparent single stars are in good agreement with newly computed predictions obtained with the binary population synthesis code BPASS and earlier estimations obtained in previous studies.