By a Higgs bundle over a formal 1d disc we mean an $n\times n$ matrix whose entries are elements of $\mathbb{C}[[x]]$, the ring of power series in one variable. Two Higgs bundles are isomorphic if they are conjugate via an element of $\operatorname{GL}(n,\mathbb{C}[[x]])$. Fixing a spectral curve (i.e. characteristic polynomial), we ask how many isomorphism classes of Higgs bundles with such spectral curve there are, and what they look like. It is well known that if the spectral curve is smooth, then there is only one isomorphism class with such spectral curve. We study in detail the case when the spectral curve has an ADE singularity. We use results from the literature on Cohen-Macaulay modules over one dimensional Cohen-Macaulay rings.