Figurative language and figures of speech, such as metaphors and hyperboles, are used every day in written and oral communication among human beings. Nonetheless, this imaginative use of words in a non literal way requires a solid understanding of semantics and a deep real-world knowledge. In the longstanding debate about whether Neural Language Models (NLMs) really have a full understanding of text, analysing how they can recognise figurative language can provide some intuition of their functioning, their capabilities and their limits. Therefore, in this paper, we exploit probing tasks to study how several NLMs of different sizes recognise four different figures of speech: hyperboles, metaphors, oxymorons and pleonasms. We analyse whether this information is learned and how it is acquired during the training of the model, describing its learning trajectory. Moreover, we analyse which layers have a better comprehension of figurative language and the influence of pre-training data. Datasets and code are available at https://github.com/nicolarici/learning-trajectories.