Rumors and information spreading emerge naturally from human-to-human interactions and have a growing impact on our everyday life due to increasing and faster access to information, whether trustworthy or not. A popular mathematical model for spreading rumors, data, or news is the Maki–Thompson model. Mean-field approximations suggested that this model does not have a phase transition, with rumors always reaching a fraction of the population. Conversely, here, we show that a continuous phase transition is present in this model. Moreover, we explore a modified version of the Maki–Thompson model that includes a forgetting mechanism, changing the Markov chain’s nature and allowing us to use a plethora of analytic and numeric methods. Particularly, we characterize the subcritical behavior, where the lifespan of a rumor increases as the spreading rate drops, following a power-law relationship. Our findings show that the dynamic behavior of rumor models is much richer than shown in previous investigations.