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I conducted an optical search for supernovae in a sample of 142 starburst galaxies over the period December, 1988, to June, 1991. The sample was drawn from two sets of galaxies: a subset of the IRAS Bright Galaxy Sample (Soifer et al. 1987), selected on the basis of their high far-infrared luminosities, and galaxies exhibiting strong emission lines of hydrogen in their nuclei (Balzano 1983). All the galaxies were nearby ($z \leq 0.03$) and relatively bright ($m_B \leq 16$). I used the one-meter Nickel telescope at Lick Observatory to obtain unfiltered or R-band CCD images with a limiting magnitude of $m_R = 18$ of as many galaxies as possible on a single night at two-week intervals. Over the course of the survey, visual comparison of new and reference images revealed a total of five supernovae in the sample, each of which occurred outside its host galaxy's nucleus. Using a set of template light curves for each type of supernova, which I constructed from the literature, in both V and R bandpasses, I determine the supernova rates in the extra-nuclear regions to be $0.7h^2$ SNU for type Ia, $0.7h^2$ SNU for type Ib/c, and $\sim 0.6h^2$ SNU for type II, with large uncertainties but upper limits of $2.2h^2$, $2.5h^2$ and $1.7h^2$ SNU, respectively. These rates are similar to those measured in ``normal'' galaxies. Because supernovae in the nuclei of galaxies are difficult to detect by comparing two images by eye, I performed aperture photometry on the nucleus of each image and searched the resulting light curves for increases in brightness caused by supernovae. I found no evidence for a supernova-induced brightening in any nucleus, and, with a few reasonable assumptions, can place upper limits of $9h^2$, $12h^2$ and $8h^2$ SNU on the rates of type Ia, Ib/c and II supernovae inside the nuclei. I derive relationships between stellar population, supernova rate and flux in Balmer emission lines for a number of population models; for the 34 galaxies in the sample with measured nuclear ${\rm H} \alpha$ and ${\rm H} \beta$ fluxes, I apply the relationships to place constraints on their stellar populations. In an appendix, I describe my contributions to the Berkeley Automatic Imaging Telescope.
