Resonant ultrasound spectroscopy has been used to characterize elastic softening and acoustic dissipation behaviour in single crystal and ceramic samples of LaAlO₃ between 10 and 300 K. For the twinned single crystals, average values of the cubic elastic moduli and C₄₄ were followed while the ceramic sample provided data for the bulk and shear moduli. A Debye-like dissipation peak occurs in the vicinity of 250 K, from which an activation energy of 43 ± 6 kJ mol⁻¹ has been obtained. The mechanism for this is not known, but it is associated with C₄₄ and therefore could be related in some way to the transition at ~817 K. Slight softening in the temperature interval  K of resonance peaks determined by shear elastic moduli hints at an incipient E_g ferroelastic instability in LaAlO₃. The softening interval ends with a further dissipation peak at ~60 K, the origin of which is discussed in terms of freezing of atomic motions of La and/or Al away from their high symmetry positions in the structure. LaAlO₃ thus shows evidence of incipient structural instability at low temperatures which is potentially analogous with the phenomenologically rich behaviour of SrTiO₃.