Our discourses are full of potential lexical ambiguities, due in part to the pervasive use of words having multiple senses. Sometimes, one word may even be used in more than one sense throughout a text. But, to what extent is this true for different kinds of texts? Does the use of polysemous words change when a discourse involves two people, or when speakers have time to plan what to say? We investigate these questions by comparing the polysemy level of texts of different nature, with a focus on spontaneous spoken dialogs; unlike previous work which examines solely scripted, written, monolog-like data. We compare multiple metrics that presuppose different conceptualizations of text polysemy, i.e., they consider the observed or the potential number of senses of words, or their sense distribution in a discourse. We show that the polysemy level of texts varies greatly depending on the kind of text considered, with dialog and spoken discourses having generally a higher polysemy level than written monologs. Additionally, our results emphasize the need for relaxing the popular "one sense per discourse" hypothesis.