Using a method to discover and classify supernovae (SNe) in galaxy spectra, we detect 91 Type Ia SNe (SNe Ia) and 16 Type II SNe (SNe II) among ∼740 000 galaxies of all types and ∼215 000 star-forming galaxies without active galactic nuclei, respectively, in Data Release 9 of the Sloan Digital Sky Survey. Of these SNe, 15 SNe Ia and eight SNe II are new discoveries reported here for the first time. We use our SN samples to measure SN rates per unit mass as a function of galaxy stellar mass, star-formation rate (SFR), and specific SFR (sSFR), as derived by the MPA-JHU Galspec pipeline. We show that correlations between SN Ia and SN II rates per unit mass and galaxy stellar mass, SFR, and sSFR can be explained by a combination of the respective SN delay-time distributions (the distributions of times that elapse between the formation of a stellar population and all ensuing SNe), the ages of the surveyed galaxies, the redshifts at which they are observed, and their star formation histories. This model was first suggested by Kistler et al. for the SN Ia rate–mass correlation, but is expanded here to SNe II and to correlations with galaxy SFR and sSFR. Finally, we measure a volumetric SN II rate at redshift 0.075 of RII, V = 0.621+0.197−0.154(stat)+0.024−0.063(sys)×10−4 yr−1 Mpc−3. Assuming that SNe IIP and IIL account for 60 per cent of all core-collapse (CC) SNe, the CC SN rate is RCC, V = 1.04+0.33−0.26(stat)+0.04−0.11(sys)×10−4 yr−1 Mpc−3.