A powerful approach for dynamic optimisation in the presence of uncertainty is to incorporate measurements into the optimisation framework so as to track the optimum. For non-singular control problems, this can be done by tracking active constraints along boundary arcs and using neighbouring-extremal (NE) control along interior arcs. Essentially, NE control forces the first-order variation of the necessary conditions of optimality (NCO) to zero. In this article, an extension of NE control to singular control problems is proposed. This article focuses on single-input systems, while the extension to multiple-input systems is investigated in the companion paper. The idea is to design NE controllers from successive time differentiations of the first-order variation of the NCO. Approximate NE feedback laws are also proposed, which are both easily implementable and tractable from a real-time optimisation perspective. These developments are illustrated by the case study of a semi-batch chemical reactor.