Let $M$ be a smooth manifold, $p\in M$ and $f\in C^\infty(M\setminus\{p\})$.
We will say that $f$ has a power-law singularity at $p$ of order $\eta$ if for every smooth immersion $\gamma:(-1,1)\to M$ such that $\gamma(0)=p$ the function $H_{f,\gamma}:(0,1)\to \mathbb{R}$, $H_{f,\gamma}(t)=t^\eta f(\gamma(t))$, extends continuously to a smooth function on $[0,1)$ and $H_{f,\gamma}(0)\neq 0$.
Fix integer $r\geq 0$. We say that two functions $f,g\in C^\infty(M\setminus\{p\})$ with a power-law singularity of order $\eta$ at $p$ are equivalent if $H_{f,\gamma}-H_{g,\gamma}\in o(t^r)$. We can then study the infinite dimensional "jet space" $J^{r,\eta}_p$ defined as the space of equilvalence classes of functions with a power-law singularity of order $\eta$ at $p$.
More generally, instead of $p$ we can consider an embedded submanifold $N\subset M$, $f\in C^\infty(M\setminus N)$, and require $\gamma(0)\in N$ with a transverse intersection, but otherwise disjoint.
My question is: has this or a similar construction been studied in the literature? What are some keywords to look for? I'm especially interested in the case of a singularity along a submanifold $N$.
FWIW, I suspect that the above construction is not completely natural and one might want slightly stronger assumptions in the definition of the power-law singularity, althought I haven't studied this very carefully.
