  We say a closed point $x$ on a curve $C$ is sporadic if there are only
finitely many points on $C$ of degree at most deg$(x)$. In the case where $C$
is the modular curve $X_1(N)$, most known examples of sporadic points come from
elliptic curves with complex multiplication (CM). We seek to understand all
sporadic points on $X_1(N)$ corresponding to $\mathbb{Q}$-curves, which are
elliptic curves isogenous to their Galois conjugates. This class contains not
only all CM elliptic curves, but also any elliptic curve
$\overline{\mathbb{Q}}$-isogenous to one with a rational $j$-invariant, among
others. In this paper, we show that all non-CM $\mathbb{Q}$-curves giving rise
to a sporadic point of odd degree lie in the $\overline{\mathbb{Q}}$-isogeny
class of the elliptic curve with $j$-invariant $-140625/8$. In addition, we
show that a stronger version of this finiteness result would imply Serre's
Uniformity Conjecture.
