A fundamentally distinct feature in quantum mechanics compared to classical physics is the existence of measurements that cannot be performed simultaneously. This is known as measurement incompatibility, which is an essential resource for many quantum information-processing tasks. In this talk, I will introduce a new method for verification of joint measurability using phase-space quasiprobability distributions. This method establishes a connection between two notions of non-classicality, namely the negativity of quasiprobability distributions and measurement incompatibility. I will also discuss incompatibility-breaking sufficient conditions for bosonic systems and Gaussian channels. In particular, these conditions provide useful tools for investigating the effects of errors and imperfections on the incompatibility of measurements in practice.