How to distinguish between zeroth, second and higher order sounds? As I understand it, be it liquid helium or phonons in a solid, one can define the second sound, zeroth sound and higher order sounds as quantum mechanical effects where heat propagates as a wave. These effects are categorically different from usual diffusion. Usually related to fluctuations or excitations or quasiparticles.
However how can one distinguish zeroth from second sound experimentally? Can they also be distinguished from third and fourth sounds?
 A: All these modes are oscillations in the conserved densities (particle number, energy, momentum, etc.) of an interacting many-body system in approximate thermal equilibrium.
Consider first ordinary (first) sound. An ordinary fluid has five conserved densities, mass (particle number), energy, and momentum. The corresponding hydrodynamic theory is then describes five modes of excitation. Three of these are diffusive (non-propagating): heat, and the two transverse components of the momentum density (shear modes). The longitudinal component of the momentum density couples to mass and energy density forms a pair ($\omega=\pm k$) of sound modes that propagate with velocity $c_s^2=(\partial P)/(\partial\rho)|_{s/n}$. This sound modes is weakly damped, but the damping grows at the mean free path increases (typically, as the temperature is lowered).
At low temperature most fluids solidify, but some substances (most notably $^4He$, a boson,  and $^3He$, a fermion) remain liquid and become quantum fluids. A Bose fluid (like $^4He$) eventually becomes superfluid, and the sound mode of the normal fluid continues into the superfluid, where we can eventually understand it as a quantized excitation of the superfluid (a phonon). In a Fermi fluid (like $^3He$) ordinary sound becomes strongly damped, but there is a particle-hole mode (which we can understand as an oscillation of the Fermi surface) that has the quantum numbers of ordinary sound, and is known as zero sound. The transition is smooth, but it is apparanant from the behavior of sound velocity and attenuation.
The image below is for liquid helium three, and shows a transition from zero to first sound at approximately 10 mK.

In a superfluid there is another hydrodynamic mode, associated with the Goldstone mode $\varphi$ related to spontaneous $U(1)$ breaking. The superfluid velocity is the gradient of this field, $\vec{v}_s=\hbar\vec\nabla\varphi/m$. Excitations of this mode mix with ordinary sound, and diagonalizing the system of equations leads to two propagating modes, known as first and second sound. The first sound mode is the one that connect to ordinary sound at the critical temperature $T_c$, whereas the speed of second sound goes to zero.
In liquid helium ordinary sound is to good approximation a density wave, whereas second sound is an oscillation of the normal fluid against the superfluid in which the density is approximately constant, but the entropy density oscillates. As a result, I can excite first sound with a vibrating plate, and second sound with a pulsed heater (and then check that the velocities behave as expected).
Below is an image of the two modes in an ultracold atomic gas (top panel: first sound; lower panel: second sound). We observe that second sound is slower, and that it cannot enter the normal fluid regime (dashed line).

Finally, third and fourth sound only exist in special geometries (thin films and channels).
