Summary: The paper is a contribution to the theory of branching processes with discrete time and a general phase space in the sense of \textit{A. Liemant, K. Matthes} and \textit{A. Wakolbinger} [``Equilibrium distributions of branching processes'' (1988; Zbl 0671.60076)]. We characterize the class of regular, i.e. in a sense sufficiently random, branching processes \((\Phi_k)_{k\in \mathbb{Z}}\) by almost sure properties of their realizations without making any assumptions about stationarity or existence of moments. This enables us to classify the clans of \((\Phi_k)\) into the regular part and the completely non-regular part. It turns out that the completely non-regular branching processes are built up from single-line processes, whereas the regular ones are mixtures of left-tail trivial processes with a Poisson family structure.