This paper describes an investigation of the oscillations and the stability of a family of stellar systems which are uniformly rotating, homogeneous spheroids. The members of this family are distinguished by the shapes of their figures, the rates of their rotations, and the shapes of their velocity ellipsoids. The characteristic value problem governing the normal modes of oscillation is formulated in a Lagrangian representation, and a method of solution is described. Solutions of the characteristic value problem are obtained explicitly for seven distinct families of modes. These include counterparts of the second-harmonic modes of oscillation in the classical Maclaurin spheroids and modes that can exhibit so-called 'firehose' instabilities. The Eulerian perturbations of the gravitational potentials of these modes are polynomials of degrees 2 and 3, respectively, in the Cartesian components of the position. For each family of modes examples are presented of the dependence of the characteristic frequencies on the structures and kinematics of the equilibrium configurations, and conditions for instability are delineated. This work generalizes and extends a number of earlier investigations of small perturbations in a homogeneous spheroid of stars. The results are of interest as a guide in the investigation of the stability of more 'realistic' modelsmore » of stellar systems, as a source of exact models with which to test N-body codes, and as a guide in the Lagrangian formulation of variational and matrix methods for more general investigations of the oscillations and stability of stellar systems. 42 refs.« less