Analogue of singularity theory in other categories 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: 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 $\text{Lk}^-_f(v)$.
