# Ehresmann's theorem for singular varieties

Let $X$ and $Y$ be two algebraic varieties over $\mathbb{C}$ and $f\colon X\rightarrow Y$ be a proper map. Assume that $Y$ is smooth.

I am interested in sufficient and necessary conditions for $f$ to be topologically locally trivial on the target, i.e., whether for each $y\in Y$ there exists an open neighborhood $V$ such that $f$ factors through a homeomorphism $f^{-1}(V){\cong} V\times f^{-1}(y)\rightarrow V$.

If $X$ is a smooth variety, the answer comes from Ehresmann's theorem: $f$ is topologically locally trivial on $Y$ if and only if it is a proper submersion. This also implies that $f$ is a smooth proper algebraic map.

But what if $X$ is singular? I could not find a reference.
Surely, $f$ must be topologically locally trivial on the $source$. Then I would like to know which tools may be used to patch together these local trivializations along a fiber, and to have a reasonable understanding of when this is not possible.

Another (necessary but perhaps insufficient) condition (at least if $Y$ is the germ of a curve) is the vanishing of vanishing cycles.