Charge density waves (CDW's) exist in a few circumstances that I know of.
The simplest example I can think of is the CDW due to nesting of the Fermi surface. This can be discussed in the context of the 'Peierls instability.' Kittel's Introduction to Solid State Physics provides a decent description of the phonon renormalization in this case (pp 422). Essentially, in a 1D metal, a gap opens in the electronic bands when the symmetry is lowered by a lattice distortion with Q = twice the Fermi wave vector (the so called 'nesting vector'). The levels near the band gap shift down so that the total electronic energy is lowered slightly. However, the lattice distortion introduces some 'strain' energy in the total electronic/lattice system. So if one minimizes the total energy with respect to the lattice distortion, one finds a stable configuration with a slightly distorted lattice.
From the point of view of the electrons, there is now a state at the BZ edge with group velocity = 0 and wave vector +- the Fermi wave vector. This is a standing wave which appears as a periodic modulation of the charge density. From the point of view of the phonons, the periodically modulated charge density perturbs the lattice towards distortions at nesting vector. If you work out a lattice dynamical problem, you will find that forces that aren't restorative (i.e. an instability) lead to imaginary frequencies.
This is not a phenomena unique to Kohn-Sham formalism and is not restricted to 0 temperature, but there is a 'Peierls transition temperature' where thermal occupation of electrons to higher energy states overcomes the simple energy lowering described above and the distortion disappears.
There is a related phenomena called a 'Kohn anomaly' where (in the Lindhard approximation) the dielectric function has a divergence at the nesting vector. Essentially the ion-ion interaction is screened by the electrons, but for electrons with k = +- the Fermi wave vector, the Lindhard expression for the dielectric function blows up and the phonon energy is renormalized to 0. See Ashcroft and Mermin, chapter 26 for a better discussion.
There is another popular example of materials hosting CDW's: the cuprates. In these materials, the existence of CDW's is a contentious issue. Static CDW's have been observed in a few isolated instances (narrow doping and temperature range), but it is argued that dynamic CDW fluctuations exist more broadly in the cuprates. These CDW's are speculated to exist due to many body correlations and aren't well understood. Another issue is that observing dynamic CDW's is difficult. However, one viewpoint (taken by my research group) is to look for phonon anomalies at nesting wave vector in these materials and, by analogy to Kohn/Peierls anomalies, argue that dynamic CDW waves are present. If you are interested in this, see https://doi.org/10.1103/PhysRevB.101.184508.