  Let $S(a,b)$ denote the normalized Dedekind sum. We study the range of
possible values for $S(a,b)=\frac{k}{q}$ with $\gcd(k,q)=1$. Girstmair proved
local restrictions on $k$ depending on $q\pmod{12}$ and whether $q$ is a square
and conjectured that these are the only restrictions possible. We verify the
conjecture in the cases $q$ even, $q$ a square divisible by $3$ or $5$, and
$2\le q\le 200$ (the latter by computer), and provide progress towards a
general approach.
