We derive exact solutions to the sine-Gordon equation describing localized structures on the background of librational and rotational travelling waves. In the case of librational waves, the exact solution represents a localized spike in space-time coordinates (a rogue wave) that decays to the periodic background algebraically fast. In the case of rotational waves, the exact solution represents a kink propagating on the periodic background and decaying algebraically in the transverse direction to its propagation. These solutions model the universal patterns in the dynamics of fluxon condensates in the semi-classical limit. The different dynamics are related to modulational instability of the librational waves and modulational stability of the rotational waves.