  We show that a triangulated motivic category admits categorical Thom
isomorphisms for vector bundles with an additional structure if and only if the
generalized motivic cohomology theory represented by the tensor unit object
admits Thom classes. We also show that the stable $\mathbb{A}^1$-derived
category does not admit Thom isomorphisms for oriented vector bundles and, more
generally, for symplectic bundles. In order to do so we compute the first
homology sheaves of the motivic sphere spectrum and show that the class in the
coefficient ring of $\mathbb{A}^1$-homology corresponding to the second motivic
Hopf map $\nu$ is nonzero which provides an obstruction to the existence of a
reasonable theory of Thom classes in $\mathbb{A}^1$-cohomology.
