Depending on the position and size of large zero submatrices within structured random matrices, the associated spectral density exhibits a power law singularity at the origin, signaling the transition to non-invertibility. For centered Hermitian as well as non-Hermitian matrices with a block variance profile we provide a complete classification of these singularities. We present an explicit finite step algorithm for computing the precise power law behavior from the position of zero blocks within the matrix. To derive this algorithm we determine the spectral density in the bulk regime by solving the associated Dyson equation. Subsequently we infer the singular behavior of the density close to the origin by deriving an equation for the exponents associated to the power laws with which the resolvent entries corresponding to the individual blocks diverge to infinity or converge to zero. For matrices with a small number of blocks we determine the universal scaling limit at the singularity and compute the local spectral density on the eigenvalue spacing scale. This derivation is based on an integral representation of the Stieltjes-transform of the spectral density that is obtained using the supersymmetry method. This is joint work with David Renfrew and Markus Ebke.