To investigate the extent to which nuclear starbursts or other nuclear activity may be connected with enhanced star formation activity in the host galaxy, we perform a statistical investigation of supernovae (SNe) discovered in host galaxies from four samples: the Markarian galaxies sample, the Second Byurakan Survey (SBS) sample, the north Galactic pole ( NGP) sample of active or star-forming galaxies, and the NGP sample of normal galaxies. Forty-seven SNe in 41 Mrk galaxies, 10 SNe in six SBS galaxies, 29 SNe in 26 NGP active or star-forming galaxies, and 29 SNe in 26 NGP normal galaxies have been studied. We find that the rate of SNe, particularly core-collapse ( Types Ib/c and II) SNe, is higher in active or star-forming galaxies in comparison with normal galaxies. Active or star-forming host galaxies of SNe are generally of later morphological type and have lower luminosity and smaller linear size than normal host galaxies of SNe. The radial distribution of SNe in active and star-forming galaxies shows a higher concentration toward the center of the active host galaxy than is the case for normal host galaxies, and this effect is more pronounced for core-collapse SNe. Ib/c-type SNe have been discovered only in active and star-forming galaxies of our samples. About 78% of these SNe are associated with H ii regions or are located very close to the nuclear regions of these active galaxies, which are in turn hosting AGNs or starburst nuclei. Besides these new results, our study also supports the conclusions of several other earlier papers. We find that Type Ia SNe occur in all galaxy types, whereas core-collapse SNe of Types Ib/c and II are found only in spiral and irregular galaxies. The radial distribution of Type Ib SNe in their host galaxies is more centrally concentrated than that of Type II and Ia SNe. The radial distances of Types Ib/c and II SNe, from the nuclei of their host galaxies, is larger for barred spiral hosts. Core-collapse SNe are concentrated in spiral arms and are often close to or in the H ii regions, whereas Type Ia SNe show only a loose association with spiral arms and no clear association with H ii regions.