I will consider the random operator obtained by perturbing the Laplacian with a white noise. In dimension 1, the construction falls within a rather classical framework, while in higher dimensions the operator is singular and renormalisation is required. I will review the literature on this operator, notably on its spectral properties. The main question that drives this investigation is: what do the eigenfunctions look like? Based on joint works with Laure Dumaz and Yueh-Sheng Hsu.