We analyse the observed fractions of core-collapse supernova (SN) types from the Lick Observatory Supernova Search (LOSS), and we discuss the corresponding implications for massive star evolution. For a standard initial mass function, observed fractions of SN types cannot be reconciled with the expectations of single-star evolution. The mass range of Wolf–Rayet (WR) stars that shed their hydrogen envelopes via their own mass-loss accounts for less than half of the observed fraction of Type Ibc supernovae (SNe Ibc). The true progenitors of SNe Ibc must extend to a much lower range of initial masses than classical WR stars, and we argue that most SN Ibc and SN IIb progenitors must arise from binary Roche lobe overflow. In this scenario, SNe Ic would still trace higher initial mass and metallicity, because line-driven winds in the WR stage remove the helium layer and propel the transition from SN Ib to Ic. Less massive progenitors of SNe Ib and IIb may not be classical WR stars; they may be underluminous with weak winds, possibly hidden by overluminous mass-gainer companions that could appear as B[e] supergiants or related objects having aspherical circumstellar material. The remaining SN types (II-P, II-L and IIn) need to be redistributed across the full range of initial masses, so that even some very massive single stars retain H envelopes until explosion. We consider the possibility of direct collapse to black holes without visible SNe, but find this hypothesis difficult to accommodate in most scenarios. Major areas of remaining uncertainty are (1) the detailed influence of binary separation, rotation and metallicity; (2) mass differences in progenitors of SNe IIn compared to SNe II-L and II-P; and (3) the fraction of SNe Ic arising from single stars with the help of eruptive mass-loss, how this depends on metallicity and how it relates to diversity within the SN Ic subclass. Continued studies of progenitor stars and their environments in nearby galaxies, accounting for SN types, may eventually test these ideas.