[Disclaimer - I am one of the co-authors of the 2D database on Materials Cloud (What you call "2D structures and layered materials", publishing the data of this work: N. Mounet et al., Nature Nanotech. 13, 246–252 (2018) so I will mostly refer to it below]
In general, these studies "extract" a layer from a bulk 3D material, and then often perform a crystal-structure relaxation on the resulting 2D layer to find a local minimum. Some studies also perform additional tests to understand if the material is really exfoliable (e.g. in Mounet et al. we compute also the binding energy of the bulk layered material).
The relaxation of the structure, however, is not enough to guarantee that the system is stable. Here are just some of the possible reasons why you could end up only in a local energy minimum:
- the relaxation in many codes tries to keep the crystal symmetry, so you might remain in a high-symmetry configuration (e.g. at the top of a double well);
- in general, all these structual minimisation algorithms find only local minima close to the starting point (technically, it depends on which "basin of attraction" the starting point is);
- You might need a larger supercell to find a stable structure.
In Mounet et al., for a subset of the materials we found (easily exfoliable with small number of atoms in the unit cell, i.e. all those available on Materials Cloud) we performed an additional step (first column of page 250 of the paper), where
First, we compute phonon frequencies at the Γ point in the Brillouin zone and, if we find imaginary-frequency modes (this can happen if atoms are constrained by symmetries to be in a saddle point), we displace them along the modes' eigenvectors first (see Methods), and then fully relax the structure. The procedure is iterated until all phonon energies at the Γ point become real and positive.
Therefore, for the systems from the Materials Cloud database, you should always find all phonons positive at Γ; there might still be phonons that are imaginary out of Γ, hinting at the need of a larger supercell, and/or at charge-density waves.
For completeness, the method used to perform the algorithm above uses the algorithm of A. Togo, I. Tanaka, Phys. Rev. B 87, 184104 (2013) and it's described in mode detail in the Methods section of the paper (and, of course, in Togo's paper in full detail).
Finally, one final important note: when computing phonons of 2D materials in codes using periodic boundary conditions, it is crucial to use an appropriate correction to correct the effect of periodic images. But most importantly, the physics is quite different in 2D from 3D, for what concerns long-range effects on the phonon dispersion (just to mention one thing, there is no LO-TO splitting in 2D but a discontinuity in the derivative of the phonon branches).
The physics is described here: T. Sohier et al., Nano Lett. 17, 3758–3763 (2017) and the method is implemented in Quantum ESPRESSO (and maybe also in some other codes).
In Mounet et al. we compute phonons taking these effects into account - you can check this, for instance, in the phonon dispersion of BN
where indeed there is no LO-TO splitting for the phonons (see e.g. the phonons at Γ around 40 THz).