"sheaves of germs of differentiable functions are by no means coherent"? This is related to a remark in Iitaka's algebraic geometry sec 1.12.
"...It should be noted that sheaves of germs of differentiable functions are by no means coherent. These facts seem to suggest coherence is linked with the property of being algebraic or analytic." 
$\textbf{Q1:}$ What is the example of non-coherence for the differentiable case? First what is the sheaf of rings in the context? Is it ring of smooth functions?
$\textbf{Q2:}$ If I recall correctly, there are analytic sheaves which are not coherent.(I do not think I will recall this correctly.) Coherence is related notion to algebraic for sure but I have to use GAGA to say it is analytic. However, in analytic setting, there are non-coherent sheaves as well. Should I naively interpret coherence is subcase of analytic or algebraic?(But not the reverse in general?)
 A: I'm going to assume you're asking about whether the structure sheaf is coherent as there are easy examples of non-coherent sheaves over any space: take an infinite direct sum of structure sheaves.
In the $C^\infty$ case, the structure sheaf will not be coherent in general. Consider the sheaf $\mathcal{O}$ of $C^\infty$ functions on $\Bbb R$. Let $f:\Bbb R\to \Bbb R$ be the function which is $0$ for $x\leq 0$ and $e^{-\frac{1}{x^2}}$ otherwise. Then $\mathcal{K}=\ker(\mathcal{O}\stackrel{f\cdot}{\to}\mathcal{O})$ is not of finite type. One can see this as follows: if $\mathcal{K}$ were of finite type, then the stalk of $\mathcal{K}$ should be of finite type over the stalk of $\mathcal{O}$ at each point. If this were true at $0$, then as $x\mathcal{K}_0=\mathcal{K}_0$, we would have that $\mathcal{K}_0=0$ by Nakayama's lemma. But it's clearly nonzero - $f(-x)$ is in it, for instance.
In the analytic case, whether or not the structure sheaf is coherent depends on the base field. For complex-analytic spaces, the structure sheaf is always coherent as a result of Oka's theorem (a hard result!). For real-analytic manifolds, the structure sheaf is again coherent, essentially because we can extend the solutions of our real-analytic equations cutting out our manifold a little bit in the complex direction and retain smoothness, and then apply Oka's theorem. For real-analytic spaces, there are no guarantees, and in fact there are real-analytic spaces which have non-coherent structure sheaves.
An example of a real-analytic variety which isn't coherent is Cartan's umbrella: $$X=\{x\in\Bbb R^3\mid x_3(x_1^2+x_2^2)-x_1^3=0\}$$ We may see that the ideal $I_X\subset \mathcal{O}_{\Bbb R^3}^{an}$ is generated at the origin by $g(x)=x_3(x_1^2+x_2^2)-x_1^3$ by complexifying and applying the Nullstellensatz. On the other hand, in a neighborhood of any point $(0,0,t)\in\Bbb R^3$ with $t\neq 0$, $M$ reduces to the line $x_1=x_2=0$ and $I_X$ is generated by $x_1,x_2$. So in a neighborhood of the origin, $I_X$ cannot be generated by $g$, which implies $I_X$ is not of finite type. As $I_X$ is the kernel of the map between the structure sheaves of $\Bbb R^3$ and $X$ in the analytic topology, this provides what you're looking for.
