We study the properties of gravito-inertial waves in a differentially rotating fluid inside a spherical shell. The fluid is modeled with the Boussinesq approximation and has a shellular steady rotation profile that stems from the combined effects of stratification, rotation, and no-slip boundary conditions. The waves properties are examined by computing paths of characteristics in the non-dissipative limit, and by solving the full dissipative eigenvalue problem using a high-resolution spectral method. Gravito-inertial waves are found to obey a mixed-type second-order operator and to be often focused around short-period attractors of characteristics or trapped in a wedge formed by turning surfaces and boundaries. We also find eigenmodes that show a weak dependence with respect to viscosity and heat diffusion just like truly regular modes. Some axisymmetric modes are found unstable and likely destabilized by baroclinic instabilities. Similarly, some non-axisymmetric modes that meet a critical layer (or corotation resonance) can turn unstable at sufficiently low diffusivities. In all cases, the growth rate of the unstable modes is determined by the differential rotation. For many modes of the spectrum, neat power laws are found for the dependence of the damping rates with the diffusion coefficients, but the theoretical explanation for the exponent values remains elusive in general. These results show a very rich and complex eigenvalue spectrum which lets us suppose an even richer and more complex spectrum when realistic models of stellar and planetary set-ups are to be considered.