Space and space-time cluster detection is an important tool for public health research and practice. The predominant approach to cluster detection is spatial and space-time scan statistics, implemented in the free SaTScan software. The scan statistic approach treats cluster detection as a large-scale multiple testing problem, using Monte Carlo simulation under the null hypothesis to identify appropriate critical values to maintain global Type I error control. In this paper, we instead envision cluster detection as a large-scale variable selection problem and propose the LASSO (least absolute shrinkage and selection operator) as an appropriate tool for cluster detection. The variable selection (LASSO) approach naturally allows for multiple clusters, while the hypothesis testing approaches to incorporating multiple clusters are typically ad hoc. We evaluate the operating characteristics (false detection rates, true detection rates, accuracy of detected clusters) of the LASSO approach in a series of simulation studies. We demonstrate the application of our method to municipality-level breast cancer incidence data from three Japanese perfectures.