  In this paper we count the number $N_3^{\text{tor}}(X)$ of $3$-dimensional
algebraic tori over $\mathbb{Q}$ whose Artin conductor is bounded by $X$. We
prove that $N_3^{\text{tor}}(X) \ll_{\varepsilon} X^{1 + \frac{\log 2 +
\varepsilon}{\log \log X}}$, and this upper bound can be improved to
$N_3^{\text{tor}}(X) \ll X (\log X)^4 \log \log X$ under the Cohen-Lenstra
heuristics for $p=3$. We also prove that for $67$ out of $72$ conjugacy classes
of finite nontrivial subgroups of $\operatorname{GL}_3(\mathbb{Z})$, Malle's
conjecture for tori over $\mathbb{Q}$ holds up to a bounded power of $\log X$
under the Cohen-Lenstra heuristics for $p=3$ and Malle's conjecture for quartic
$A_4$-fields.
