  We prove that finite area isolated singularities of surfaces with constant
positive curvature in R^3 are removable singularities, branch points or
immersed conical singularities. We describe the space of immersed conical
singularities of such surfaces in terms of the class of real analytic closed
locally convex curves in the 2-sphere with admissible cusp singularities,
characterizing when the singularity is actually embedded. In the global
setting, we describe the space of peaked spheres in R^3, i.e. compact convex
surfaces of constant positive curvature with a finite number of singularities,
and give applications to harmonic maps and constant mean curvature surfaces.
