Solid state materials having crystal atomic structure appear to have a unique set of discovered quasi-particles like polaron, phonon, exiton, magnon, soliton, polariton, and some other ones - see the following link:

 https://en.wikipedia.org/wiki/List_of_quasiparticles     

Quasi-particles are responsible for numerous physical effects in crystals giving opportunities to interpret experiments qualitatively and quantitatively as well. Hence, quasi-particles are a type of a knowledge infrastructure of solid state physics.

Professor Dragoe (with whose answer above I am in a full heart agreement) already mentioned importance of phonons in studying thermal conductivity. I would mention yet significance of phonons in interpreting and calculating free carriers’ mobility dependence of temperature in metals and semiconductors.

Unfortunately, you don't give the context of what exactly you mean by "calculations". I assume therefore, that the context are DFT calculations, i.e. band structure calculations.

In some codes, the atomic positions are set fixed, in others they can be varied. Within the DFT calculation you would seek for the set of atomic coordinates which minimizes the total energy (T=0 equilibrium atomic positions).

In case experimental data are available you can compare this result with the factual lattice parameters determined for the same material by experiment. Usually one would hope that a good agreement is an argument for the chosen DFT model to be accurate.

Calculating the phonon spectrum (=lattice vibrations) can provide you with an additional check. The phonon frequencies stand for the (effective) interatomic spring constants associated with vibrational normal modes. If these turn out to be correct (in agreement with measured data [= phonon spectra]) then you have more reason to be confident about the calculation.

In a solely theoretical context, phonon calculations may indeed provide a hint towards the stability of the calculated struture. If some phonon frequency is particularly small, then this means that a distortion along this normal coordinate costs very little energy. In the extreme case, the energy curvature along some coordinate might turn out to be negative - then you know for sure that (within the chosen model) the structure is unstable. [I would think that in 2D materials not all normal modes lend themselves to such an analysis, since the TA phonon energy in the limit k->0 becomes arbitrarily small {if I am wrong here, would someone correct me, please}].

Otherwise, knowing the phonon spectrum, one can also compute the phonon contribution to specific heat, which could yet be another basic material property of interest.

In conjunction with symmetry analysis, you can determine which phonons are Raman or IR active and which ones are "silent" modes.

So there are quite a few reasons why one could be interested in computing the vibrational properties... [just my $0.02]

acoustic phonons cause structural instability. there are many examples in oxides and their themionical stability