Wave equation
I think the question is vaguely posed, since the answer depends on what we define as waves and wave equations. In the question cited in the OP many answers simply assumed that waves mean electromagnetic waves and wave equations means
$$
\partial_t^2u(\mathbf{x},t)=c^2\nabla^2u(\mathbf{x},t).$$
The dispersion relation in this case is obvious:
$$\omega^2-c^2\mathbf{k}^2=0.$$
Linear equations
One could talk about waves in more general sense, as solutions to any linear equation, solvable via Fourier transform, i.e., having solutions
$$
u(\mathbf{x},t) =\int d\mathbf{k}\int d\omega \tilde{u}(\mathbf{k},\omega)e^{i(\mathbf{k}\mathbf{x}-\omega t)},$$
in which case any linear operator would suffice
$$F(\partial_t, \nabla)u(\mathbf{x},t)=0.$$
By choosing function $F(\partial_t, \nabla)$ one could get almost anything. E.g.,
$$\partial_t^4u(\mathbf{x},t)=a\nabla^8u(\mathbf{x},t) + \nabla^4u(\mathbf{x},t) + cu(\mathbf{x},t)$$
has several dispersion branches.
Among more basic equations with several branches one could cite Dirac equation and Klein-Gordon equation (the latter being simply the wave equation with a constant term added).
Non-linear equations
One could go even further and consider non-linear equations that allow running solutions of the type $$f(\mathbf{k}\mathbf{x}-\omega t),$$ such as, e.g., Korteveg-de Vries equation or Sine-Gordon equation.
Which of these equations do happen?
In university physics courses one typically deals with linear theories, because the fundamental phsyics is described (mainly?) by linear theories. In more domain-specific courses one however quickly encounters equations that have higher derivatives or non-linear terms. The domains to look for more complex equations are:
- hydrodynamics
- elasticity theory
- electrodynamics of non-linear media
- non-linear theory (which deals more specifically with the equations rather than their physical content).
Remarks
First-order equations One can have also first-order wave equations, e.g.,
$$\partial_t u(x,t)\pm v\partial u(x,t)=0,$$
which give actial travelling wave solutions of type $f(x-vt)$. In more dimensions:
$$\partial_t u(\mathbf{x},t)-\mathbf{v}\cdot\nabla u(\mathbf{x},t)=0.$$
The nuance of these equations is taht they have a preferred direction for the wave propagation (even in 1D we have either right- or left-moving wave, depending on the sign). This is why the physical theories that are symmetric in space and/or time usually have second (or generally even) partial derivatives.
One example of an equation with such a first-order term is the Navier-Stokes equation, although it is a non-linear one (but it can be linearized to give simple wave solutions).
Waves vs. running waves When dealing with general form of equation $F(\partial_t, \nabla)u(\mathbf{x},t)=0$, it is necessary to keep in mind that, although it is solvable by Fourier transform, its solutions are not necessarily running waves of the form $f(\omega t-\mathbf{k}\mathbf{x})$. Requiring that solutions have this form would restrict the type of the differential operators that can be used, e.g., excluding diffusion equation.
Schrödinger equation On the other hand, Schrödinger equation (which can be viewed as a diffusion equation with complex coefficients) is certainly considered a wave equation and its solutions are often referred to as matter waves, even though they are not running waves in the restricted sense mentioned above.
Broad/flat band limit in some solid state phsyics problems one considers a broad-band limit where all electrons are assumed to have the same wave function (or wave number), while possibly ahving different energies - this can be interpreted as a continuum of frequencies corresponding to the same wavelength. The opposite and also used is the flat-band limit, where one assumes that all the wave numbers correspond to the same energy/frequency.
Best Answer
The wave mechanics dispersion relation you cite is for EM waves propagating in free space. In other media, the dispersion relation is not necessarily linear (it can be quadratic or have some more complex dependence). So in this context, there's nothing special about quantum mechanics.
More generally, the dispersion relation tells us about the phase speed of the wave and the group velocity:
$$v_\text{phase} = \frac{\omega_n}{k_n}$$
and
$$ v_\text{g} \equiv \frac{\partial \omega_n}{\partial k_n} $$
So for example, in contrast to EM waves in free space, the particular quantum dispersion relation you cite will have a group velocity that depends on the wavenumber. The quantum mechanics interpretation of this is that the particle's momentum will depend on its wavenumber ($p = \hbar k$).