Residual displacements are sensitive to ground motion details, hence more random than peak inelastic displacements. Among the factors with systematic impact on residual displacements, the post-yield-stiffness-ratio has been studied thoroughly; its effects are not investigated further. Concerning another important factor, the hysteresis law, past studies have focused on the bilinear model, which does not represent concrete structures. Residual displacements from nonlinear response-history analyses of bilinear systems are compared to those from models tuned to concrete structures, conforming to modern codes, deficient or intermediate. Deficient-type structures, with their narrow, almost self-centering hysteresis loops, develop markedly smaller residual displacements than those with stable energy-dissipating behavior. A velocity pulse in the motion increases peak inelastic and residual displacements by about the same proportion. As a fraction of the peak inelastic or spectral displacement, residual displacements are on average almost independent of the period and increase when the lateral strength ratio increases, reaching a limit at a lateral strength ratio of 2 to 5. Peak inelastic displacements are a better basis for estimation of residual displacements than spectral ones: the ratio of the two is almost independent of the period, the lateral strength ratio (beyond values of 2 to 3) and velocity pulses. The spectrum of the ratio of residual displacement to peak inelastic or spectral displacement is considered as a random process of period; its mean and variance functions, marginal probability distributions and autocorrelation functions are given in terms of the lateral strength ratio, the hysteresis model and the presence of a velocity pulse.