  The study of the geometry of $n$-uniform measures in $\mathbb{R}^{d}$ has
been an important question in many fields of analysis since Preiss' seminal
proof of the rectifiability of measures with positive and finite density. The
classification of uniform measures remains an open question to this day. In
fact there is only one known example of a non-trivial uniform measure, namely
$3$-Hausdorff measure restricted to the Kowalski-Preiss cone. Using this cone
one can construct an $n$-uniform measure whose singular set has Hausdorff
dimension $n-3$. In this paper, we prove that this is the largest the singular
set can be. Namely, the Hausdorff dimension of the singular set of any
$n$-uniform measure is at most $n-3$.
