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Magic-angle twisted multilayer graphene hosts many correlated phases, including correlated insulators, unconventional superconductivity, magnetism and non-trivial topology. In the case of twisted trilayer graphene, flat-bands emerge at the magic-angle condition – which is given by the ratio between the two twist angles. Depending on the degree of lattice relaxation, either polycrystals or quasicrystals emerge. While superconductivity and magnetism have been observed in transport experiments, no local measurements have been reported up to date. Here we present low-temperature Scanning Tunneling Microscopy and Spectroscopy studies on twisted trilayer graphene. We observe a smoothly varying moiré that corresponds to the quasicrystal case. We observe a series of gapped phases when doping both to the electron and hole sides. These gaps have a non-trivial response to magnetic field and a strong spatial dependence, and match the doping and electric field regions where superconductivity has been found.