  Magnomechanical devices, in which magnetic excitations couple to mechanical
vibrations, have been discussed as efficient and broadband microwave signal
transducers in the classical and quantum limit. We experimentally investigate
the magnetoelastic coupling between the ferromagnetic resonance (FMR) modes in
a metallic Co$_{25}$Fe$_{75}$ thin film, featuring ultra-low magnetic damping
as well as sizable magnetostriction, and standing transverse elastic phonon
modes in sapphire, silicon and gadolinium gallium garnet by performing
broadband FMR spectroscopy at cryogenic temperatures. For all these substrate
materials, we observe an interaction between the resonant acoustic and magnetic
modes, which can be tailored by the propagation direction of the acoustic mode
with respect to the crystallographic axes. We identify these phonon modes as
transverse shear waves propagating with slightly different velocities with
relative magnitudes of $\Delta v/v\simeq10^{-5}$, i.e., all substrates show
phononic birefringence. Upon appropriately choosing the phononic mode, the
hybrid magnomechanical system enters the Purcell enhanced coupling regime.
