Elastic and anelastic properties of single crystal samples of EuTiO₃ have been measured between 10 and 300 K by resonant ultrasound spectroscopy at frequencies in the vicinity of 1 MHz. Softening of the shear elastic constants C44 and 1/2(C₁₁−C₁₂) by ∼20–30% occurs with falling temperature in a narrow interval through the transition point, Tc=284 K, for the cubic-tetragonal transition. This is accounted for by classical coupling of macroscopic spontaneous strains with the tilt order parameter in the same manner as occurs in SrTiO₃. A peak in the acoustic loss occurs a few degrees below Tc and is interpreted in terms of initially mobile ferroelastic twin walls, which rapidly become pinned with further lowering of temperature. This contrasts with the properties of twin walls in SrTiO₃, which remain mobile down to at least 15 K. No further anomalies were observed that might be indicative of strain coupling to any additional phase transitions above 10 K. A slight anomaly in the shear elastic constants, independent of frequency and without any associated acoustic loss, was found at ∼140 K. It marks a change from elastic stiffening to softening with falling temperature and perhaps provides evidence for coupling between strain and local fluctuations of dipoles related to the incipient ferroelectric transition. An increase in acoustic loss below ∼80 K is attributed to the development of dynamical magnetic clustering ahead of the known antiferromagnetic ordering transition at ∼5.5 K. Detection of these elastic anomalies serves to emphasize that coupling of strain with tilting, ferroelectric, and magnetic order parameters is likely to be a permeating influence in determining the structure, stability, properties, and behavior of EuTiO₃.