  The paper discusses stably trivial torsors for spin and orthogonal groups
over smooth affine schemes over infinite perfect fields of characteristic
unequal to 2. We give a complete description of all the invariants relevant for
the classification of such objects over schemes of dimension at most $3$, along
with many examples. The results are based on the
$\mathbb{A}^1$-representability theorem for torsors and transfer of known
computations of $\mathbb{A}^1$-homotopy sheaves along the sporadic isomorphisms
to spin groups.
