A good reference for this is the paper 
> Y. Katznelson and K. Stromberg. *"Everywhere differentiable, nowhere monotone, functions*, Am. Math. Monthly, **81 (4)**, (1974), 349-353.

There, the authors give an explicit construction of a function $L:\mathbb R\to\mathbb R$ that is differentiable, with bounded derivative, and such that all three sets $\{x\mid L'(x)>0\}$, $\{x\mid L'(x)=0\}$, and $\{x\mid L'(x)<0\}$ are dense in $\mathbb R$. The argument is flexible enough that we can arrange $\{x\mid L'(x)=0\}=\mathbb Q$, if wanted. The function $L$ is the difference of two monotone functions, and $L'$ is not Riemannn integrable over any interval. 

In the paper, they indicate that the first example of a differentiable, nowhere monotone, function is due to Köpke, in 1889, with a further example due to Pereno in 1897. (I have not examined their constructions myself, so I do not know whether there are mistakes in their presentation, as the other answer suggests, but Pereno's function is discussed as an example in Hobson's book from 1957.) 

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For this kind of results, two obvious references to consult are **Counterexamples in Analysis**, by Gelbaum and Olmsted, and **A second course on real functions**, by van Rooij and Schikhof. The latter presents the Katznelson-Stromberg construction in detail.