Whitney, Thom, Mather, Arnold and others develoved the singularity theory of smooth maps. Does there exist any analogue of this theory in the category of TOP or PL (or Lipschitz) maps? I mean notions like: stability, hierarchy of singularities, nice dimensions, arising stratification on the source and on the target of a map, according to the germ type of the map at the points, Thom polynomials.

A brief account of PL Morse theory has been requested by Andras in the comments, so I am writing it down here. Note that this does not address the main question on PL singularity theory. Note also that a much better account of PL Morse theory can be found in Mladen Bestvina's notes. Let $K$ be a polyhedral cell complex and denote the attaching map of each $n$simplex $\sigma$ of $K$ by $\phi_\sigma:\Delta^n \to K$ where $\Delta^n$ is the standard $n$simplex. Def. A function $f:K \to \mathbb{R}$ is PL Morse if (a) $f\circ \phi_\sigma:\Delta^n \to \mathbb{R}$ is constant only for $n=0$, affine for $n \geq 1$, and (b) the image under $f$ of the vertices forms a discrete subset of $\mathbb{R}$. This definition sets things up so that the only candidates for critical points of $f:K \to \mathbb{R}$ are the vertices of $K$. For each vertex $v$, the descending link $\text{Lk}^_f(v)$ consists of all those simplices $\sigma$ not containing $v$ such that $v \cup \sigma =: \sigma'$ is a simplex of $K$ and $f_{\sigma'}$ attains its maximum on $v$. The descending link plays the role of stable manifold in PL Morse theory. Here's the main Theorem: Thm. If $I = [a,b]$ is an interval so that the only vertex in $f^{1}(I)$ is $v$ with $f(v) = b$, then $f^{1}(I)$ is homotopy equivalent rel $f^{1}(a)$ to $f^{1}(a)$ attached with the cone on $\text{Lk}^_f(v)$. 

