Rotation is a key parameter in the evolution of massive stars, affecting their evolution, chemical yields, ionizing photon budget, and final fate. We determined the projected rotational velocity, υ e sin i, of ~330 O-type objects, i.e. ~210 spectroscopic single stars and ~110 primaries in binary systems, in the Tarantula nebula or 30 Doradus (30 Dor) region. The observations were taken using VLT/FLAMES and constitute the largest homogeneous dataset of multi-epoch spectroscopy of O-type stars currently available. The most distinctive feature of the υ e sin i distributions of the presumed-single stars and primaries in 30 Dor is a low-velocity peak at around 100 km s-1. Stellar winds are not expected to have spun-down the bulk of the stars significantly since their arrival on the main sequence and therefore the peak in the single star sample is likely to represent the outcome of the formation process. Whereas the spin distribution of presumed-single stars shows a well developed tail of stars rotating more rapidly than 300 km s-1, the sample of primaries does not feature such a high-velocity tail. The tail of the presumed-single star distribution is attributed for the most part - and could potentially be completely due - to spun-up binary products that appear as single stars or that have merged. This would be consistent with the lack of such post-interaction products in the binary sample, that is expected to be dominated by pre-interaction systems. The peak in this distribution is broader and is shifted toward somewhat higher spin rates compared to the distribution of presumed-single stars. Systems displaying large radial velocity variations, typical for short period systems, appear mostly responsible for these differences.