We compute the type Ia, Ib/c and II supernova (SN) rates as functions of the cosmic time for galaxies of different morphological types. We use four different chemical evolution models, each one reproducing the features of a particular morphological type: E/S0, S0a/b, Sbc/d and Irr galaxies. We essentially describe the Hubble sequence by means of decreasing efficiency of star formation and increasing infall timescale. These models are used to study the evolution of the SN rates per unit luminosity and per unit mass as functions of cosmic time and as functions of the Hubble type. Our results indicate that: (i) the observed increase of the SN rate per unit luminosity and unit mass from early to late galaxy types is accounted for by our models. Our explanation of this effect is related to the fact that the latest Hubble types have the highest star formation rate per unit mass; (ii) By adopting a Scalo (1986) initial mass function in spiral disks, we find that massive single stars ending their lives as Wolf-Rayet objects are not sufficient to account for the observed type Ib/c SN rate per unit mass. Less massive stars in close binary systems can give instead a significant contribution to the local Ib/c SN rates. On the other hand, with the assumption of a Salpeter (1955) IMF for all galaxy types, single massive WR stars are sufficient to account for the observed type Ib/c SN rate. (iii) Our models allow us to reproduce the observed type Ia SN rate density up to redshift z~1. We predict an increasing type Ia SN rate density with redshift, reaching a peak at redshift z >= 3, because of the contribution of massive spheroids. ; Comment: ApJ, accepted for publication. 17 pages, 11 figures