I'd just like to add something about the plot of the graph of the function $z=\mathrm{f}(x,y)$.
The term "singularity", in this context, does not refer to a function whose graph is singular.
The Giant Rat is a function germ $\mathrm{f} : (\mathbb{R}^2,0) \to (\mathbb{R},0)$. We're interested in the set of $(x,y) \in \mathbb{R}^2$, very close to $(0,0) \in \mathbb{R}^2$ which are sent to $0 \in \mathbb{R}$. In other words, what does $$\mathrm{f}^{-1}(0) = \{(x,y) \in \mathbb{R} : \mathrm{f}(x,y)=0\}$$ look like in a small neighbourhood of the origin?
If $a \neq 0,1$ then $xy(x-y)(x-ay)=0$ are four lines all crossing at $(0,0)$. This is quite a nasty singularity. Consider, for a moment, the $D_4$ singularity $x^3-xy^2$. The zero-level set of this is three distinct real lines through the origin. Any function germ $\mathrm{g}:(\mathbb{R}^2,0) \to (\mathbb{R},0)$, whose degree three Taylor series is a cubic polynomial in $x$ and $y$ whose zero-level set is three distinct real lines through the origin is $\mathscr{R}$-equivalent to $x^3-xy^2$, i.e. there is a diffeomorphism $\mathrm{\phi} : (\mathbb{R}^2,0) \to (\mathbb{R}^2,0)$ for which $$\mathrm{g} \circ \phi = x^3 - xy^2$$ This isn't totally unexpected because any three concurrent lines can be taken to any other three concurrent lines via a linear transformation, meaning that any cubic polynomial whose zero-level set is three distinct real lines can be taken to $x^3 - xy^2$ by a linear transformation. Going back to the Giant Rat, with $a \neq 0,1$, we have four distinct real lines through the origin. Four concurrent lines can be taken to four other concurrent lines by a projective transformation if, and only if, they have the same cross ratio. The Giant Rat is so nasty because there are an uncountable number of singularity types in the family $xy(x-y)(x-ay)$ which are not $\mathscr{R}$-equivalent. The zero-level sets $xy(x-y)(x-ay)=0$ are almost all different from each other in the sense that no diffeomorphism can take $xy(x-y)(x-ay)=0$ to $xy(x-y)(x-by)$, where $a \neq b$.