Detailed spectral classifications are presented for 352 O-B0 stars in the VLT-FLAMES Tarantula Survey ESO Large Programme, of which 213 O-type are judged of sufficiently high quality for further morphological analysis. Among them, six subcategories of special interest are distinguished. (1) Several new examples of the earliest spectral types O2-O3 have been found, while a previously known example has been determined to belong to the nitrogen-rich ON2 class. (2) A group of extremely rapidly rotating main-sequence objects has been isolated, including the largest vsini values known, the spatial and radial-velocity distributions of which suggest ejection from the two principal ionizing clusters NGC 2070 and NGC 2060. (3) Several new examples of the evolved, rapidly rotating Onfp class show similar evidence, although at least some of them are spectroscopic binaries. (4) No fewer than 48 members of the Vz category, hypothesized to be on or near the zero-age main sequence, are found in this sample; in contrast to the rapid rotators, they are strongly concentrated to the ionizing clusters and a newly recognized region of current and recent star formation to the north, supporting their interpretation as very young objects, as do their relatively faint absolute magnitudes. (5) A surprisingly large fraction of the main-sequence spectra belong to the recently recognized V((fc)) class, with CIII emission lines of similar strength to the usual NIII in V((f)) spectra, although a comparable number of the latter are also present, as well as six objects with very high-quality data but no trace of either emission feature, presenting new challenges to physical interpretations. (6) Two mid-O Vz and three late-O giant/supergiant spectra with morphologically enhanced nitrogen lines have been detected. Absolute visual magnitudes have been derived for each star with individual extinction laws, and composite Hertzsprung-Russell diagrams provide evidence of the multiple generations present in this field. Spectroscopic binaries, resolved visual multiples, and possible associations with X-ray sources are noted. Astrophysical and dynamical analyses of this unique dataset underway will provide new insights into the evolution of massive stars and starburst clusters.