Most types of supernovae (SNe) have yet to be connected with their progenitor stellar systems. Here, we reanalyze the 10-year SN sample collected during 1998-2008 by the Lick Observatory Supernova Search (LOSS) in order to constrain the progenitors of SNe Ia and stripped-envelope SNe (SE SNe, i.e., SNe~IIb, Ib, Ic, and broad-lined Ic). We matched the LOSS galaxy sample with spectroscopy from the Sloan Digital Sky Survey and measured SN rates as a function of galaxy stellar mass, specific star formation rate, and oxygen abundance (metallicity). We find significant correlations between the SN rates and all three galaxy properties. The SN Ia correlations are consistent with other measurements, as well as with our previous explanation of these measurements in the form of a combination of the SN Ia delay-time distribution and the correlation between galaxy mass and age. The ratio between the SE SN and SN II rates declines significantly in low-mass galaxies. This rules out single stars as SE~SN progenitors, and is consistent with predictions from binary-system progenitor models. Using well-known galaxy scaling relations, any correlation between the rates and one of the galaxy properties examined here can be expressed as a correlation with the other two. These redundant correlations preclude us from establishing causality - that is, from ascertaining which of the galaxy properties (or their combination) is the physical driver for the difference between the SE SN and SN II rates. We outline several methods that have the potential to overcome this problem in future works.