  Let $a \in \mathbb{Z}_{>0}$ and $\epsilon_1, \epsilon_2, \epsilon_3 \in \{\pm
1\}$. We classify explicitly all singular moduli $x_1, x_2, x_3$ satisfying
either $\epsilon_1 x_1^a + \epsilon_2 x_2^a + \epsilon_3 x_3^a \in \mathbb{Q}$
or $(x_1^{\epsilon_1} x_2^{\epsilon_2} x_3^{\epsilon_3})^{a} \in
\mathbb{Q}^{\times}$. In particular, we show that all the solutions in singular
moduli $x_1, x_2, x_3$ to the Fermat equations $x_1^a + x_2^a + x_3^a= 0$ and
$x_1^a + x_2^a - x_3^a= 0$ satisfy $x_1 x_2 x_3 = 0$. Our proofs use a
generalisation of a result of Faye and Riffaut on the fields generated by sums
and products of two singular moduli, which we also establish.
