It seems that there is some confusion here, starting form the question.

As I understand, Dipayan Sen has run two lattice-dynamical calculations for a same material, once using the elementary cell (EC), once using a supercell (SC, i.e. a multiple of the EC), both of them at the Gamma point of the respective unit cell. Did I get it right, Dipayan?

The unit cell of a material is NOT unique, Alexander: you can always choose any multiple of the primitive cell. This is the very concept of "supercell".

I am also a bit confused by Andrew's answer, which seems a bit contradictory to me. The second paragraph of his answer is substantially correct. Each point of the Brillouin Zone (BZ) of a SC displays as many modes as those of the elementary cell, times the number of ECs contained in the SC. This is so because some finite k points of the EC BZ are "folded" into the Gamma point of the SC. If you find a soft mode at the Gamma point of the SC that was not present at the Gamma point of the EC, this simply means that a mode at a finite k of the EC BZ that has been folded into the SC Gamma point has gone soft. This may have nothing to do with numerical issues (hence, the first paragraph of Andrew's answer may result to be a bit confusing). To make sure, check that the frequencies that you compute at Gamma for the EC are also present among those that you compute for the SC. Hope this helps.

It depends precisely on what you mean by "a particular unit cell" and "different unit cell".

For example, do you mean using different sized supercells (for example 1x1x1 or 2x2x2 in the first case and then 3x3x3 in the second case). If you had negative modes in the first case but not the second case, it could be due to numerical issues. Generally gamma point modes can be captured in primitive cell, but may require a large number of k-points (sometimes this is resolved if when you go to a supercell, the k-point grid wasn't fully scaled down).

If you mean that one point group phase has soft modes (imaginary frequencies) at Gamma and another does not, then this means that the structure is dynamically unstable.  It is similar to having a soft mode at an arbitrary high symmetry point, except that it means the primitive cell itself can accommodate the symmetry lowering displacement.  When the soft mode occurs at an arbitrary high symmetry point (for example the X point in cubic HfO2)m it requires enlargement of the cell (a 2x1x1 or 1x2x1 or 1x1x2) for the symmetry lowering displacement to be accommodated.

It seems that there is some confusion here, starting form the question.

As I understand, Dipayan Sen has run two lattice-dynamical calculations for a same material, once using the elementary cell (EC), once using a supercell (SC, i.e. a multiple of the EC), both of them at the Gamma point of the respective unit cell. Did I get it right, Dipayan?

The unit cell of a material is NOT unique, Alexander: you can always choose any multiple of the primitive cell. This is the very concept of "supercell".

I am also a bit confused by Andrew's answer, which seems a bit contradictory to me. The second paragraph of his answer is substantially correct. Each point of the Brillouin Zone (BZ) of a SC displays as many modes as those of the elementary cell, times the number of ECs contained in the SC. This is so because some finite k points of the EC BZ are "folded" into the Gamma point of the SC. If you find a soft mode at the Gamma point of the SC that was not present at the Gamma point of the EC, this simply means that a mode at a finite k of the EC BZ that has been folded into the SC Gamma point has gone soft. This may have nothing to do with numerical issues (hence, the first paragraph of Andrew's answer may result to be a bit confusing). To make sure, check that the frequencies that you compute at Gamma for the EC are also present among those that you compute for the SC. Hope this helps.