Passive quasi-one-dimensional random media are known to exhibit one of the three regimes of transport - ballistic, diffusive or localized - depending on the system size. In contrast, in non-conservative systems, the physical parameter space also includes the gain/absorption length scale. Here, by studying the relationships between the transport mean free path, the localization length, and the gain/absorption length, we enumerate 15 regimes of wave propagation through quasi-one-dimensional random media with gain or absorption. The results are presented graphically in the form of a phase diagram. Of particular experimental importance in an absorbing random medium, we identify three different regimes that bear the signatures of the localized regime of the passive counterpart. We also review the literature and, when possible, assign experimental systems to a particular regime on the diagram.