A family of classical superintegrable Hamiltonians, depending on an arbitrary
radial function, which are defined on the 3D spherical, Euclidean and
hyperbolic spaces as well as on the (2+1)D anti-de Sitter, Minkowskian and de
Sitter spacetimes is constructed. Such systems admit three integrals of the
motion (besides the Hamiltonian) which are explicitly given in terms of ambient
and geodesic polar coordinates. The resulting expressions cover the six spaces
in a unified way as these are parametrized by two contraction parameters that
govern the curvature and the signature of the metric on each space. Next two
maximally superintegrable Hamiltonians are identified within the initial
superintegrable family by finding the remaining constant of the motion. The
former potential is the superposition of a (curved) central harmonic oscillator
with other three oscillators or centrifugal barriers (depending on each
specific space), so that this generalizes the Smorodinsky-Winternitz system.
The latter one is a superposition of the Kepler-Coulomb potential with another
two oscillators or centrifugal barriers. As a byproduct, the Laplace-Runge-Lenz
vector for these spaces is deduced. Furthermore both potentials are analysed in
detail for each particular space. Some comments on their generalization to
arbitrary dimension are also presented.