We investigate the class of nonnegative potentials V(x) for which the Schrödinger equation −Δ u+V u=0 admits a unique type of singular solution such that u(x)→∞ as x→0. This class includes the potentials with inverse-square growth at 0, i.e. 0≤ V(x)≤ C| x|. If for instance we fix boundary data u=g at | x|=1 then the singular solution is unique up to a multiplicative factor.