  Fix an alphabet $A=\{0,1,\dots,M\}$ with $M\in\mathbb{N}$. The univoque set
$\mathscr{U}$ of bases $q\in(1,M+1)$ in which the number $1$ has a unique
expansion over the alphabet $A$ has been well studied. It has Lebesgue measure
zero but Hausdorff dimension one. This paper investigates how the set
$\mathscr{U}$ is distributed over the interval $(1,M+1)$ by determining the
limit $$f(q):=\lim_{\delta\to
0}\dim_H\big(\mathscr{U}\cap(q-\delta,q+\delta)\big)$$ for all $q\in(1,M+1)$.
We show in particular that $f(q)>0$ if and only if
$q\in\overline{\mathscr{U}}\backslash\mathscr{C}$, where $\mathscr{C}$ is an
uncountable set of Hausdorff dimension zero, and $f$ is continuous at those
(and only those) points where it vanishes. Furthermore, we introduce a
countable family of pairwise disjoint subsets of $\mathscr{U}$ called {\emph
relative bifurcation sets}, and use them to give an explicit expression for the
Hausdorff dimension of the intersection of $\mathscr{U}$ with any interval,
answering a question of Kalle et al.~[{\emph arXiv:1612.07982; to appear in
Acta Arithmetica}, 2018]. Finally, the methods developed in this paper are used
to give a complete answer to a question of the first author [{\emph Adv.
Math.}, 308:575--598, 2017] about strongly univoque sets.
