The growth of single hydrogen bubbles at micro-electrodes is studied in an acidic electrolyte over a wide range of concentrations and cathodic potentials. New bubble growth regimes have been identified which differ in terms of whether the bubble evolution proceeds in the presence of a monotonic or oscillatory variation in the electric current and a carpet of microbubbles underneath the bubble. Key features such as the growth law of the bubble radius, the dynamics of the microbubble carpet, the onset time of the oscillations and the oscillation frequencies have been characterized as a function of the concentration and electric potential. Furthermore, the system's response to jumps in the cathodic potential has been studied. Based on the analysis of the forces involved and their scaling with the concentration, potential and electric current, a sound hypothesis is formulated regarding the mechanisms underlying the micro-bubble carpet and oscillations.