  The tetradymite compound BiSbTeSe$_2$, is one of the most bulk-insulating
three-dimensional topological insulators, which makes it important in the
topological insulator research. It is a member of the solid-solution system
Bi$_{2-x}$Sb$_x$Te$_{3-y}$Se$_y$, for which the local crystal structure, such
as the occupation probabilities of each atomic site, is not well understood. We
have investigated the temperature and polarization dependent spontaneous Raman
scattering in BiSbTeSe$_2$, revealing a much higher number of lattice
vibrational modes than predicted by group theoretical considerations for the
space group R$\bar{3}$m corresponding to an ideally random solid-solution
situation. The density functional calculations of phonon frequencies show a
very good agreement with experimental data for parent material Bi$_2$Te$_3$,
where no disorder effects were found. In comparison to Bi$_2$Te$_3$ the
stacking disorder in BiSbTeSe$_2$, causes a discrepancy between theory and
experiment. Combined analysis of experimental Raman spectra and DFT calculated
phonon spectra for different types of atomic orders showed coexistence of
different sequences of layers in the material and that those with Se in the
center and a local order of Se-Bi-Se-Sb-Te, are the most favored.
