In this paper we present an overview of analytical results and numerical methods for singular boundary value problems for ordinary differential equations with a singularity of the first kind. Special attention is paid to the analysis of shooting methods, where the associated initial value problems are solved by the acceleration technique known as Iterated Defect Correction (IDeC) based on the backward Euler method, and on direct discretization using collocation schemes. Convergence, error estimation and mesh selection are discussed for both approaches. Moreover, we study the fixed point convergence of the IDeC iteration, where the fixed point corresponds to a collocation solution.