  The Snyder-de Sitter (SdS) model is a generalization of the Snyder model to a
spacetime background of constant curvature. It is an example of noncommutative
spacetime admitting two fundamental scales besides the speed of light, and is
invariant under the action of the de Sitter group. Here we consider its
nonrelativistic counterpart, i.e. the Snyder model restricted to a
three-dimensional sphere, and the related model obtained by considering the
anti-Snyder model on a pseudosphere, that we call anti-Snyder-de Sitter (aSdS).
  By means of a nonlinear transformation relating the SdS phase space variables
to canonical ones, we are able to investigate the classical and the quantum
mechanics of a free particle and of an oscillator in this framework. As in
their flat space limit, the SdS and aSdS models exhibit rather different
properties. In the SdS case, a lower bound on the localization in position and
momentum space arises, which is not present in the aSdS model. In the aSdS
case, instead, a specific combination of position and momentum coordinates
cannot exceed a constant value.
  We explicitly solve the classical and the quantum equations for the motion of
the free particle and of the harmonic oscillator. In both the SdS and aSdS
cases, the frequency of the harmonic oscillator acquires a dependence on the
energy.
