We consider second-order ergodic Mean-Field Games systems in RN with coercive potential and aggregating nonlocal coupling, defined in terms of a Riesz interaction kernel. Equilibria solve a system of PDEs where a Hamilton-Jacobi-Bellman equation is combined with a Kolmogorov-Fokker-Planck equation for the mass distribution. Due to the interplay between the strength of the attractive term and the behavior of the diffusive part, we will obtain three different regimes for existence and nonexistence of classical solutions to the MFG system. In the Hardy-Littlewood-Sobolev-supercritical regime, by means of a Pohozaev-type identity, we prove nonexistence of regular solutions to the MFG system without potential term. On the other hand, in the Hardy-Littlewood-Sobolev-subcritical regime, using a fixed point argument, we show existence of classical solutions at least for masses smaller than a given threshold value. In the mass-subcritical regime, we show that actually this threshold can be taken to be +∞. Finally, considering the MFG system with a small parameter ε > 0 in front of the Laplacian, we study the behavior of solutions in the vanishing viscosity limit, namely when the diffusion becomes negligible. First, we obtain existence of classical solutions to potential free MFG systems with Riesz-type coupling. Secondly, we prove concentration of mass around the minima of the potential.