In a number of transient electroanalytical experiments (such as, in particular, the potential step chronoamperometry) one expects current-time responses having a singularity at time t = 0. A reliable simulation of such singular responses is a difficult task. Conventional simulation techniques fail to provide accurate results close to the singularity. A recent extension [L. K. Bieniasz, J. Comput. Appl. Math. 323 (2017) 136] of the adaptive Huber method for solving Volterra integral equations (lEs) is shown to overcome this difficulty in cases when the current behaves like r(-1/2) for t -> 0. The method requires an availability of highly accurate approximants for computing certain integrals of the kernel functions occurring in the IEs. Relevant approximants are elaborated for several most important kernel functions specific of one-dimensional diffusion, and one kernel function for two-dimensional diffusion. The resulting algorithm is tested on two examples of single IEs describing chronoamperometry and cyclic voltammetry for a single reversible charge transfer, and one example of an IE system describing chronoamperometry for an ErevErev mechanism. Singular transients are simulated automatically with a prescribed accuracy, even for t arbitrarily close to the singularity.