  We consider substitutions on compact alphabets and provide sufficient
conditions for the diffraction to be pure point, absolutely continuous and
singular continuous. This allows one to construct examples for which the
Koopman operator on the associated function space has specific spectral
components. For abelian bijective substitutions, we provide a dichotomy result
regarding the spectral type of the diffraction. We also provide the first
example of a substitution that has countably infinite Lebesgue spectral
components and countably infinite singular continuous components. Lastly, we
give a non-constant length substitution on a countably infinite alphabet that
gives rise to substitutive Delone sets of infinite type. This extends the
spectral theory of substitutions on finite alphabets and Delone sets of finite
type with inflation symmetry.
