(Quasi)coherent sheaves on smooth manifolds, and their applications

Coherence is a useless notion on a differential manifold!
The reason is that there are no non trivial coherent sheaves on a differential manifold of dimension $n\gt 0$, because the structural sheaf itself $\mathcal E=\mathcal C^\infty$is not coherent.
Let me show this for the simplest manifold: $\mathbb R$.
Of course $\mathcal E$ is of finite type over itself but $\mathcal E$ is not coherent because the kernel of a sheaf morphism $\phi:\mathcal E\to \mathcal E$ is not always of finite type .
For example let $f\in \mathcal E (\mathbb R)$ be the infamous Cauchy smooth function such that $f(x)=0$ for $x\leq 0$ and $f(x)=\exp(-\frac {1}{x^2})\gt 0$ for $x\gt 0$ and consider the sheaf morphism $\phi:\mathcal E\to \mathcal E$ given by multiplication with $f$.
The kernel $\mathcal K=\operatorname {Ker} \phi\subset \mathcal E$ of $\phi$ is the ideal sheaf of smooth functions $g$ such that $g(x)=0$ for $x\gt0$ and that sheaf is not of finite type, because even the stalk $\mathcal E_0$ is not a module of finite type over the ring $\mathcal O_0$.
(Reason : $\mathcal E_0= x\mathcal E_0$ by Hadamard and finite generation would imply $\mathcal E_0=0$ by Nakayama ).

OK, but what are coherent sheaves good for anyway?
They are tremendously useful for calculating cohomology: for example they are acyclic on Stein holomorphic manifolds or affine algebraic varieties and have finite-dimensional cohomology on compact holomorphic manifolds or projective algebraic varieties.

So all is lost?
Not at all! Luckily, thanks to the existence of smooth partitions of unity on paracompact manifolds, all sheaves of finitely generated $\mathcal E$-Modules are acyclic and thus are just as useful as coherent sheaves.