Toeplitz matrices represent a wide range of non-selfadjoint matrices that have been widely studied in the past. These matrices are associated with functions, which are called symbols. Although having a particular structure, their lack of self-adjointness causes a strong spectral instability. Under some assumptions, the addition of random noise gives a "regularization of the spectrum", i.e., an equidistribution of the eigenvalues of the perturbed matrix in the numerical range of its symbol. In this talk, I will give a generalized construction of Toeplitz matrices associated with rough symbols (especially by the presence of jumps), and for which this phenomenon still holds. The analysis of the spectrum of these matrices will be done through their empirical spectral measure, and their study is based on semiclassical analysis tools.