Oscillator networks, including neuronal ensembles, can exhibit multiple cooperative rhythms such as chimera and cluster states. However, understanding which rhythm prevails remains challenging. In this talk, we address this fundamental question in the context of Kuramoto-Sakaguchi networks of rotators with higher-order Fourier modes in the coupling. We show that three-cluster splay states with two distinct, coherent clusters and a solitary oscillator are the prevalent rhythms in networks with an odd number of units. We denote such tripod patterns cyclops states with the solitary oscillator reminiscent of the Cyclops' eye. As their mythological counterparts, the cyclops states are giants that dominate the system's phase space in weakly repulsive networks with first-order coupling. Astonishingly, adding the second or third harmonics to the Kuramoto coupling function makes the Cyclops states global attractors practically across the full range of coupling's repulsion. We also extend our analysis to understand the mechanisms responsible for destroying the cyclops states and inducing new dynamical patterns called breathing and switching cyclops' states. Beyond the Kuramoto oscillators, we show that cyclops states are robustly present in networks of canonical theta-neurons with adaptive coupling. More generally, our results suggest clues for finding dominant rhythms in physical and biological networks.