  Fe$_{1.057(7)}$Te undergoes a first-order tetragonal to monoclinc structural
transition at T$_{S} \sim 70$ K, breaking the C$_{4}$ lattice symmetry and
simultaneously breaking time reversal symmetry with bicollinear magnetic order.
We investigate the soft acoustic lattice dynamics near this combined
magneto-structural transition. We apply spherically neutron polarimetry to
study the static magnetism near this transition, characterized with x-ray
powder diffraction, and find no evidence of static incommensurate magnetic
correlations near the onset of monoclinic and bicollinear antiferromagnetic
order. This fixes the position of our single crystal sample in the Fe$_{1+x}$Te
phase diagram in the magnetic bicollinear region and illustrates that our
sample statically undergoes a transition from a paramagnetic phase to a
low-temperature bicollinear phase. We then apply neutron spectroscopy to study
the acoustic phonons, related to elastic deformations of the lattice. We find a
temperature dependent soft acoustic branch for phonons propagating along [010]
and polarized along [100]. The slope of this acoustic phonon branch is
sensitive to the elastic constant $C_{66}$ and the shear modulus. The
temperature dependence of this branch displays a softening with a minimum near
the magneto-structural transition of T$_{S}$ $\sim$ 70 K and a recovery within
the magnetically ordered low temperature phase. Soft acoustic instabilities are
present in the collinear phases of the chalcogenides Fe$_{1+x}$Te where nematic
order found in Fe$_{1+\delta}$Se is absent. We speculate, based on localized
single-ion magnetism, that the relative energy scale of magnetic spin-orbital
coupling on the Fe$^{2+}$ transition metal ion is important for the presence of
a nematicity in the chalcogenides.
