This text presents some notes on the classification of very simple rules in one and two dimensions. We focus on the 256 Elementary Cellular Automata and the 64 totalistic rules with von Neumann neighbourhood. The updating scheme is fully asynchronous updating, that is, only one cell is updated randomly and uniformly at each time step. While a partial classification of such rules has been proposed some years ago, very few progress has been made since then. In order to understand the obstacles to a complete classification, we here mainly adopt an experimental approach based on the observations of the space-time diagrams. We also give some sketch of proofs when possible and underline the interesting open problems.