It is generally believed that, when a linguistic item acquires a new meaning, its overall frequency of use in the language rises with time with an S-shaped growth curve. Yet, this claim has only been supported by a limited number of case studies. In this paper, we provide the first corpus-based quantitative confirmation of the genericity of the S-curve in language change. Moreover, we uncover another generic pattern, a latency phase of variable duration preceding the S-growth, during which the frequency of use of the semantically expanding word remains low and more or less constant. We also propose a usage-based model of language change supported by cognitive considerations, which predicts that both phases, the latency and the fast S-growth, take place. The driving mechanism is a stochastic dynamics, a random walk in the space of frequency of use. The underlying deterministic dynamics highlights the role of a control parameter, the strength of the cognitive impetus governing the onset of change, which tunes the system at the vicinity of a saddle-node bifurcation. In the neighborhood of the critical point, the latency phase corresponds to the diffusion time over the critical region, and the S-growth to the fast convergence that follows. The duration of the two phases is computed as specific first passage times of the random walk process, leading to distributions that fit well the ones extracted from our dataset. We argue that our results are not specific to the studied corpus, but apply to semantic change in general.