Borel measure supported on $\mathbb{Q}$ Let $\mu$ be a Borel measure supported on $\mathbb{Q} \subset \mathbb{R}$. Must $\mu$ be a sum of Dirac measures?

Yes, because by definition of the support (depending on your definition), you have $\mu ( \Bbb{R} \setminus \Bbb{Q}) \leq \mu(\Bbb{R} \setminus {\rm supp}(mu)) = 0$ and thus for every measurable set $M$:
$$
\mu(M) = \mu(M \cap \Bbb{Q}) = \sum_{q \in \Bbb{Q} \cap M} \mu({q}) = (\sum_{q \in \Bbb{Q}} \mu({q}) \delta_q ) (M),
$$
so that $\mu$ is a (countable, nonnegative) linear combination of Dirac measures. 
If you meant a finite sum, this is not correct, take e.g. $\sum_n 1/n^2 \delta_{1/n}$. 
