We investigate, via the dynamic programming approach, a finite fuel nonlinear singular stochastic control problem of Bolza type. We prove that the associated value function is continuous and that its continuous extension to the closure of the domain coincides with the value function of a nonsingular control problem, for which we prove the existence of an optimal control. Moreover, such a continuous extension is characterized as the unique viscosity solution of a quasivariational inequality with suitable boundary conditions of mixed type.