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Since a smooth (real) manifold is canonically a locally ringed space, we can define (quasi)coherent sheaves over smooth manifolds in the usual manner. But is the category of (quasi)coherent sheaves on a smooth manifold well-behaved? That is, it an abelian category? The nLab article on the topic states that for many ringed spaces the category of coherent sheaves will fail to be abelian.

If (quasi)coherent sheaves on manifolds are indeed well-behaved, are there any interesting applications of these sheaves in differential geometry and differential topology? Obviously vector bundles, being locally free, are quasicoherent, so I'm sure there must be some nice uses of (quasi)coherent sheaves.

I'm coming to sheaf theory from a differential geometry background, so I apologize if this is obvious stuff.

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    No, vector bundles are not coherent, not even trivial ones ! Why do you think it obvious that they are? – 2017-02-27
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    Sorry, I meant quasicoherent, since being locally free implies that they are locally presentable. But why do they fail to be coherent? I've never worked with coherent modules so my intuition for them is lacking. – 2017-02-27
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    OK, I have written an answer explaining non-coherence. And, by the way, coherence is an extremely subtle concept, introduced by giants like Oka and Henri Cartan. It takes quite a while for us mere mortals to build an intuition for that concept, so don't be demoralized by your present lack of it :-) – 2017-02-27

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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.