Research software has been categorized in different contexts to serve different goals. We start with a look at what research software is, before we discuss the purpose of research software categories. We propose a multi-dimensional categorization of research software. We present a template for characterizing such categories. As selected dimensions, we present our proposed role-based, readiness-based, developer-based, and dissemination-based categories. Since our work has been inspired by various previous efforts to categorize research software, we discuss them as related works. We characterize all these categories via the previously introduced template, to enable a systematic comparison. We report on the multi-dimensional categorization of selected research software examples.