Stellar mergers are responsible for a wide variety of phenomena such as rejuvenated blue stragglers, highly magnetised stars, spectacular transients, iconic nebulae, and stars with peculiar surface chemical abundances and rotation rates. Before stars merge, they enter a contact phase. Here, we investigate which initial binary-star configurations lead to contact and classical common-envelope (CE) phases and assess the likelihood of a subsequent merger. To this end, we computed a grid of about 6000 detailed one-dimensional binary evolution models with initial component masses of 0.5-20.0M_{sun} at solar metallicity. Both components were evolved, and rotation and tides were taken into account. We identified five mechanisms that lead to contact and mergers: runaway mass transfer, mass loss through the outer Lagrange point L2, expansion of the accretor, orbital decay because of tides, and non-conservative mass transfer. At least 40 percent of mass-transferring binaries with initial primary-star masses of 5-20M{sun} evolve into a contact phase; >12 percent and >19 percent likely merge and evolve into a CE phase, respectively. Because of the non-conservative mass transfer in our models, classical CE evolution from late Case-B and Case-C binaries is only found for initial mass ratios q_i<0.15-0.35. For larger mass ratios, we find stable mass transfer. In early Case-B binaries, contact occurs for initial mass ratios q_i_<0.15-0.35, while in Case-A mass transfer, this is the case for all q_i in binaries with the initially closest orbits and q_i_<0.35 for initially wider binaries. Our models predict that most Case-A binaries with mass ratios of q0.5, confirming our model expectations. Because of non-conservative mass transfer with mass transfer efficiencies of 15-65 percent, 5-25 percent, and 25-50 percent in Case-A, -B, and -C mass transfer, respectively (for primary-star masses above 3M_{sun}_), our contact, merger, and classical CE incidence rates are conservative lower limits. With more conservative mass transfer, these incidences would increase. Moreover, in most binaries, the non-accreted mass cannot be ejected, raising the question of the further evolution of such systems. The non-accreted mass may settle into circumstellar and circumbinary disks, but could also lead to further contact systems and mergers. Overall, contact binaries are a frequent and fascinating result of binary mass transfer of which the exact outcomes still remain to be understood and explored further.