The results of recent experimental and theoretical investigations of the spectral densities of fluctuations (SDFs) of noise-driven nonlinear dynamical systems are reviewed. Emphasis is placed on the analysis of the shapes and intensities of peaks in the SDFs. Three different types of phenomena are considered. First, the SDFs of a class of monostable underdamped nonlinear systems, in which the variation of eigenfrequency with energy is nonmonotonic, are investigated. It is shown that they exhibit zero-dispersion peaks and noise-induced spectral narrowing, as well as zero-frequency peaks. Secondly, it is demonstrated that systems bistable in an external periodic field can exhibit supernarrow spectral peaks within the range of a kinetic phase transition. Finally, recent results in stochastic resonance (SR) are reviewed, including phase shifts, giant nonlinearities for weak noise, SR for periodically modulated noise intensity, and high-frequency SR for periodic attractors.