  We present an argument due to Thom to formulate a priori cohomology
obstructions for a projective variety to admit an embedded resolution of
singularities, and generalize the argument to a field of characteristic $p >
0$. We show that these obstructions are defined for a general cohomology theory
that satisfies localization, \'etale excision, and $\mathbb{A}^1$-invariance,
and is equipped with a proper pushforward. As examples, we show that odd degree
Steenrod homology operations on higher Chow groups mod $\ell$, defined by a
suitable $E_{\infty}$-algebra structure, vanish on the fundamental class of a
smooth variety, where $\ell$ is any prime. We extend this general vanishing
result to arbitrary varieties for operations defined mod $\ell \neq p$. Along
the way, we also obtain a Wu formula for the mod $p$ Steenrod operations in the
case of closed embeddings of smooth varieties.
