Phonons in non-crystalline media

Consider a solid rod made of a glassy substance, and model it as a set of atoms in random locations, held together by randomly oriented harmonic springs (enough so that the graph is rigid). The Hamiltonian of this system is in principle diagonalizable, and since all the springs are harmonic, the potential energy is quadratic in all the atomic coordinates, so the whole system is equivalent to a set of uncoupled harmonic oscillators, which are of course the normal modes. The lowest frequency mode will be rod flexing back and forth with two nodes, the second will be the second overtone with three nodes, and so on, but importantly this set of normal modes goes all the way up to modes with a number of nodes on the order of the number of atoms in the rod. None of these normal modes has a precisely defined wavevector, because of the lack of periodicity in the glass, but they do have approximate wavevectors, and of course they have precise frequencies corresponding to the eigenvalues of the Hamiltonian.

One weird thing about this that seems completely different from the usual treatment of phonons in crystals is that all these modes are standing waves - they don't propagate in a particular direction, and their approximate wavevectors are only defined up to sign. This is actually the case for a finite-size crystal as well - no propagating waves can actually be eigenstates, and instead the boundary conditions at the ends of the crystal cause them to mix into standing waves that are the exact physical eigenstates. The only reason we introduce periodic boundary conditions and talk about the propagating waves in crystals is that it's so much more convenient. Of course, if you create a wave packet at one end of the material, either crystal or glass, you can always express that in whatever basis of eigenstates you want, and as they evolve the packet will end up moving through the crystal and spreading out according to some dispersion relation.

I don't know how you would actually calculate that dispersion relation for a glass (other than brute-force computation), but it's possible in principle.

The same considerations also apply to quasicrystals, but with the interesting addition that there are now diffusively propagating modes called phasons with long relaxation times. 2

Phonons in a gas is a really weird thing to think about because in an ideal gas, the particles are assumed to be non-interacting, and they have to be in a thermal distribution of single-particle quantum states for it to be a gas (rather than a Bose-Einstein condensate or something). If the gas particles are delocalized, and don't interact with each other, then what the heck is a phonon? Yet sound waves in a gas obviously exist, so the question remains whether they're quantized or not. I can't answer this part of the question.

Phonons were named after photons and have the same functionality quantum mechanically. They are both waves and particles. Nobody disputes the wave nature of sound in general. In non ordered materials there is no way that a sound wave will give all its energy to an atom, or a cluster of atoms, as an example. The reason is because all these atoms are in an incoherent state and no pure quantum mechanical state function can be defined. An atom can have a pure state function but the wavelengths of sound are way larger than the wavelengths that an atom can absorb.

In ordered materials like crystals this can happen because phases are defined so that there can be large dimensional coherent scattering: the order allows a quantum mechanical state function for the crystal to be defined, which can interact with a sound wave so that the whole energy of that sound is absorbed by the crystal, thus a phonon.

Edit: In the comments to the question it becomes clear that there is a confusion on the use of the term "phonon". I am using the definition in wikipedia.

Edit2: Copying from Carl's comment in the question, I would add that the quantization is the familiar E=hν=ℏω, and that this applies to fluids as well. But without a translational symmetry, this cannot display quantization but instead can take any value of ω.

If we expand the definition of a phonon to a continuous spectrum, it seems that the answer to the question above finally is yes, yes, yes. Though I guess that in disordered media the particle nature is not manifest. Actually this is also true about photons in ambient light, as an example. Maybe somebody should expand the Wikipedia article.