LaAlO3 and SrTiO3 are insulating, non-magnetic oxides, yet the interface between them exhibits a two-dimensional electron system with high electron mobility(1), superconductivity at low temperatures(2-6) and electric-field-tuned metal-insulator and superconductor-insulator phase transitions(3,6-8). Bulk magnetization and magnetoresistance measurements also indicate some form of magnetism depending on preparation conditions(5,9-11) and a tendency towards nanoscale electronic phase separation(10). Here we use local imaging of the magnetization and magnetic susceptibility to directly observe a landscape of ferromagnetism, paramagnetism and superconductivity. We find submicrometre patches of ferromagnetism in a uniform background of paramagnetism, with a non-uniform, weak diamagnetic superconducting susceptibility at low temperature. These results demonstrate the existence of nanoscale phase separation as indicated by theoretical predictions based on nearly degenerate interface sub-bands associated with the Ti orbitals(12,13). The magnitude and temperature dependence of the paramagnetic response indicate that the vast majority of the electrons at the interface are localized(14), and do not contribute to transport measurements(3,6,7). In addition to the implications for magnetism, the existence of a two-dimensional superconductor at an interface with highly broken inversion symmetry and a ferromagnetic landscape in the background indicates the potential for exotic superconducting phenomena.