This paper presents methods for characterizing nonlinearities and sudden disturbances in stationary/transient responses by decomposing signals using the Hilbert-Huang transform (HHT) and a sliding-window fitting (SWF) technique. Similar to the wavelet transform SWF uses windowed regular harmonics and their orthogonality to extract local harmonic components. However, SWF decomposes a signal into less components because it allows distorted harmonics, and it provides time-varying amplitudes and frequencies of extracted components that can reveal system's nonlinearities. To extract components from a signal HHT uses the apparent time scales shown by the local maxima and minima of the signal (instead of using orthogonality of chosen fitting functions) and cubic spline fitting of extrema to sequentially sift components of different time scales, starting from highfrequency ones to low-frequency ones. Because it does not use orthogonality of functions, HHT provides more accurate time-varying amplitudes and frequencies of extracted components for accurate estimation of system characteristics and nonlinearities. Because the first extracted component contains all original discontinuities, its time-varying amplitude and frequency are excellent indicators of sudden transient disturbances. However, the discontinuity-induced Gibbs' phenomenon makes HHT analysis inaccurate around the two data ends. On the other hand, the SWF analysis has no Gibbs' phenomenon at the two data ends, but it cannot extract accurate modulation frequencies due to the use of non-orthogonal basic functions in the sliding-window least-squares curve-fitting process. Numerical and experimental results show that HHT can provide accurate extraction of intrawave amplitude- and phase-modulation, distorted harmonic response under a single-frequency harmonic excitation, softening and hardening effects, different orders of nonlinearity, interwave amplitude- and phasemodulation, multiple-mode vibrations caused by internal/external resonances, and instants of impact loading on a structure from stationary/transient responses. These phenomena are keys for performing dynamics-based structural damage detection and health monitoring.