Regular sequences are natural generalisations of fixed points of constant-length substitutions on finite alphabets, that is, of automatic sequences. Using the harmonic analysis of measures associated with substitutions as motivation, we study the limiting asymptotics of regular sequences by constructing a measure-theoretic framework surrounding them. The constructed measures are generalisations of mass distributions supported on attractors of iterated function systems. It turns our that these measures are $D$-ergodic, thus have pure spectral type. We provide results towards a classification of the spectral type based on inequalities involving the (sum) spectral radius and the joint spectral radius of the underlying finite set of matrices. The trichotomy of spectral type offers a way to view regular sequences alternative to the standard classification of the diffeo-algebraic character of generating functions.