Hierarchical Powell-Sabin splines are C1-continuous piecewise quadratic polynomials defined on a hierarchical triangulation. The mesh is obtained by partitioning an initial conforming triangulation locally with a triadic split, so that it is no longer conforming. We propose a normalized quasi-hierarchical basis for this spline space. The basis functions have a local support, they form a convex partition of unity, and they admit local subdivision. We show that the basis is strongly stable on uniform hierarchical triangulations. We consider two applications: data fitting and surface modelling.