I was surprised the first time I learned that a quintic plane curve can have an $A_{10}$ singularity i.e $x^2+y^{10}$. I am wondering if there is something about that phenomenon: Given a singularity on its normal form, and a fixed degree $d$. Is there a standard way to find an hypersurface of degree $d$ on $\mathbb{P}^n$ with the given singularity? Is there a lower bound for $d$ in terms of the singularity invariants?

For example: Given the singularity $x^2+y^3+z^{13}$ and $d=5$, the surface $$ (x+z^3)^2+(y-z^2)^3+x^3y^2+x^5 $$ has that singularity at $(0,0,0)$. How can I define a surface for another singularity e.g. $x^2+y^4+z^{22}$ ?

I am mostly thinking in surfaces and plane curves. Thanks for any hint or suggestion!