Resonant ultrasound spectroscopy has been used to characterize elastic softening and anelastic dissipation processes associated with the transition in single crystal and ceramic samples of LaAlO₃. Softening of the cubic structure ahead of the transition point is not accompanied by an increase in dissipation but follows different temperature dependences for the bulk modulus, , and the shear components, and C₄₄, as if the tilting instability contains two slightly different critical temperatures. The transition itself is marked by the complete disappearance of resonance peaks (superattenuation), which then reappear below ~700 K in spectra from single crystals. Comparisons with low frequency, high stress data from the literature indicate that the dissipation is not due to macroscopic displacement of needle twins. An alternative mechanism, local bowing of twin walls under low dynamic stress, is postulated. Pinning of the walls with respect to this displacement process occurs below ~350 K. Anelasticity maps, analogous to plastic deformation mechanism maps, are proposed to display dispersion relations and temperature/frequency/stress fields for different twin wall related dissipation mechanisms. These allow comparisons to be made of anelastic loss mechanisms under mechanical stress with elastic behaviour observed by means of Brillouin scattering at high frequencies which might also be related to microstructure.