In this paper, we study the Besov regularity of Lévy white noises on the $d$-dimensional torus. Due to their rough sample paths, the white noises that we consider are defined as generalized stochastic fields. We, initially, obtain regularity results for general Lévy white noises. Then, we focus on two subclasses of noises: compound Poisson and symmetric-$α$-stable (including Gaussian), for which we make more precise statements. Before measuring regularity, we show that the question is well-posed; we prove that Besov spaces are in the cylindrical $σ$-field of the space of generalized functions. These results pave the way to the characterization of the $n$-term wavelet approximation properties of stochastic processes.