This is the third paper of a series in which we present new measurements of the observed rates of supernovae (SNe) in the local Universe, determined from the Lick Observatory Supernova Search (LOSS). We have considered a sample of ∼1000 SNe and used an optimal subsample of 726 SNe (274 SNe Ia, 116 SNe Ibc and 324 SNe II) to determine our rates. We study the trend of the rates as a function of a few quantities available for our galaxy sample, such as luminosity in the B and K bands, stellar mass and morphological class. We discuss different choices (SN samples, input SN luminosity functions, inclination correction factors) and their effect on the rates and their uncertainties. A comparison between our SN rates and the published measurements shows that they are consistent with each other to within the uncertainties when the rate calculations are done in the same manner. Nevertheless, our data demonstrate that the rates cannot be adequately described by a single parameter using either galaxy Hubble types or B−K colours. A secondary parameter in galaxy ‘size’, expressed by luminosity or stellar mass, is needed to adequately describe the rates in the rate–size relation: the galaxies of smaller sizes have higher SN rates per unit mass or per unit luminosity. The trends of the SN rates in galaxies of different Hubble types and colours are discussed. We examine possible causes for the rate–size relation. Physically, such a relation for the core-collapse SNe is probably linked to the correlation between the specific star-formation rate and the galaxy sizes, but it is not clear whether the same link can be established for SNe Ia. We discuss the two-component (‘tardy’ and ‘prompt’) model for SN Ia rates, and find that the SN Ia rates in young stellar populations might have a strong correlation with the core-collapse SN rates. We derive volumetric rates for the different SN types [e.g. for SNe Ia, a rate of (0.301 ± 0.062) × 10−4 SN Mpc−3 yr−1 at redshift 0] and compare them to the measurements at different redshifts. Finally, we estimate the SN rate for the Milky Way Galaxy to be 2.84 ± 0.60 SNe per century (with a systematic uncertainty of a factor of ∼2), consistent with published SN rates based on several different techniques.