I understand the question as requesting applications of the geometry of spin manifolds outside of pure mathematics. There are many, starting with the Dirac equation for the spinor of a relativistic electron (1920). Recent applications center on topological states of matter and on quantum theories of gravity.
 
One recent overview is given in
<A HREF="https://arxiv.org/abs/1801.06988">Spin Geometry and Some Applications</A>.

For application beyond physics, see <A HREF="https://www.researchgate.net/publication/260225057_Spin_Geometry_and_Image_Processing">Spin Geometry and Image Processing.</A>