We examine theoretically the signatures of magnetic adatoms in graphene probed by scanning tunneling spectroscopy (STS). When the adatom hybridizes equally with the two graphene sublattices, the broadening of the local adatom level is anomalous and can scale with the cube of the energy. In contrast to ordinary metal surfaces, the adatom local moment can be suppressed by the proximity of the probing scanning tip. We propose that the dependence of the tunneling conductance on the distance between the tip and the adatom can provide a clear signature for the presence of local magnetic moments. We also show that tunneling conductance can distinguish whether the adatom is located on top of a carbon atom or in the center of a honeycomb hexagon.