Ammonium sulphate (NH4)2SO4 (AS) has recently been shown to display giant barocaloric and electrocaloric effects when its paraelectric-ferrielectric transition at ∼225 K is driven by hydrostatic pressure or an electric field, respectively. Important aspects of the structural evolution in this context are coupling of strain to molecular librational modes, distortions, and rotations of NH4+ and SO42− groups, and to changes in hydrogen bonding. Resonant ultrasound spectroscopy (RUS) has been used to reveal subtle elastic and anelastic relaxation processes of single crystals of AS through the temperature interval 5–300 K that are indicative of (i) coupling of strain to the driving order parameter, modified by a transformation microstructure, and (ii) of hysteretic effects that are attributed to different dynamics of displacive and order/disorder components of the phase transition. At high temperatures, partial chemical decomposition results in the formation of triammonium hydrogen disulphate (NH4)3H(SO4)2 (AHS) which displays elastic and anelastic anomalies associated with phase transitions at ∼265, 139, 132, and 72 K. Freezing of defects coupled with strain occurs at temperatures below ∼35 K in both AS and AHS.