In *Singular points of complex hypersurfaces*, John Milnor proves the following theorem:

Let $x \in V$ be a point on a variety $V$ in $\mathbb{R}^n$ or $\mathbb{C}^n$. Assume $x$ is either a smooth point or an isolated singularity. Let $D_{\epsilon}$ be the closed $\epsilon$-ball about $x$, $S_{\epsilon}$ its boundary (the sphere about $x$ of radius $\epsilon$), and $K = V \cap S_{\epsilon}$. Then for $\epsilon$ sufficiently small, the pair $(D_{\epsilon}, V \cap D_{\epsilon})$ is homeomorphic to the pair $(CS_{\epsilon}, CK)$, where $C$ denotes taking the cone. (Theorem 2.10)

In Remark 2.11, Milnor observes that this theorem "likely" holds even if $x$ is a non-isolated singularity; in particular, it is known even in this case that "a suitably chosen neighborhood of any point is homeomorphic to the cone over something."

This book was written in 1968. What is the current status of this problem?

something. The question is whether that "something" is $K$. – Charles Staats Jul 24 '10 at 15:07elsehas written it up. Because it's a folk semi-result: People can guess that it's true and suggest approaches to prove it, but it's still messy and maybe no one has done the work. Theorems of this type often don't have a status, and are often proven as intermediate steps. – Greg Kuperberg Jul 24 '10 at 22:28