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In this paper we study various fractal geometric aspects of the Minkowskiquestion mark function $Q.$ We show that the unit interval can be written asthe union of the three sets $\Lambda_{0}:=\{x:Q'(x)=0\}$,$\Lambda_{\infty}:=\{x:Q'(x)=\infty\}$, and $\Lambda_{\sim}:=\{x:Q'(x)$ doesnot exist and $Q'(x)\not=\infty\}.$ The main result is that the Hausdorffdimensions of these sets are related in the following way.$\dim_{H}(\nu_{F})<\dim_{H}(\Lambda_{\sim})= \dim_{H} (\Lambda_{\infty}) =\dim_{H} (\mathcal{L}(h_{\mathrm{top}}))<\dim_{H}(\Lambda_{0})=1.$  Here, $\mathcal{L}(h_{\mathrm{top}})$ refers to the level set of the  Stern-Brocot multifractal decomposition at the topological entropy  $h_{\mathrm{top}}=\log2$ of the Farey map $F,$ and $\dim_{H}(\nu_{F})$denotes the Hausdorff dimension of the measure of maximal entropy of thedynamical system associated with $F.$ The proofs rely partially on themultifractal formalism for Stern-Brocot intervals and give non-trivialapplications of this formalism.

Fractal analysis for sets of non-differentiability of Minkowski's question mark function