Abstract In this work well-posedness and stability properties of the abstract spline problem are studied in the framework of reflexive spaces. Tykhonov well-posedness is proved without restrictive assumptions. In the context of Hilbert spaces, also the stronger notion of Levitin–Polyak well-posedness is established. A sequence of parametric problems converging to the given abstract spline problem is considered in order to study stability. Under natural assumptions, convergence results for sequences of solutions of the perturbed problems are obtained.