One way to analyse aperiodic systems employs spectral notions, either via dynamical systems theory or via harmonic analysis. In this talk, we will look at two particular aspects of this, after a quick overview of how the diffraction measure can be used for this purpose. First, we consider some consequences of inflation rules on the spectra via renormalisation, and how to use it to exclude absolutely continuous components. Second, we take a look at a class of dynamical systems of number-theoretic origin, how they fit into the spectral picture, and what (other) methods there are to distinguish them.