  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
about 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 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 and Milky Way rates ... (abridged)
