Lattice instability and soft phonons in single-crystal<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">La</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">Sr</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mi mathvariant="normal">Cu</mml:mi><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">O</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>

The dispersion of the low-lying phonon branches of several doped and undoped single crystals of ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$ has been investigated by using inelastic-neutron-scattering techniques. The zone-center modes are in reasonable agreement with Raman measurements. The reported peaks in the phonon density of states show up at energies that correspond to extrema in the dispersion curves of the transverse and longitudinal acoustic branches near the zone boundary. The tetragonal-to-orthorhombic phase transition is caused by a softening of a transverse-optic-phonon mode at the $X$ point. The rotational nature of the soft mode leads to moderate electron-phonon coupling and the mode is unlikely to enhance significantly conventional phonon mediated superconductivity. We did not observe any evidence for the predicted breathing-mode instability near the zone boundary.