Suppose $X$ is a $k$-variety of dimension $d$, and suppose $TX$ is its tangent bundle. Consider the (triangulated, stable $\infty$-,...) categories of perfect complexes $\text{Perf}(X)$ and $\text{Perf}(TX)$ on $X$ and $TX$, respectively. Is it true that even if $X$ is singular (but nice enough if necessary), then *locally* $\text{Perf}(TX)$ is equivalent to $\text{Perf}(U\times\mathbb{A}^d)$, $U$ open subscheme of $X$?

This question is motivated by the following. If $X$ is smooth, then the tangent bundle is locally an $\mathbb{A}^d$-bundle. I want a notion similar to this that works for singular schemes. Is there an obstruction that measures this defect? Going back to perfect complexes, "how far" is $\text{Perf}(TX)$ from $\text{Perf}(X)$?

Any pointers to the appropriate literature would be nice.