Serre famously proved that for elliptic curves $E$ over number fields $k$ without complex multiplication, the galois group $H$ of the field generated over $k$ by all the torsion points $E_{\text{tor}}$ of $E$ is a subgroup of finite index in $G=\displaystyle\lim_{\leftarrow\atop n} \text{GL}_2(\Bbb Z/n\Bbb Z)$. When $k=\Bbb Q$, the smallest the index of $H$ in $G$ can be is 2, and if it is, we say $E$ is a Serre curve over $\Bbb Q$. Now let $E$ be an elliptic curve over $\Bbb Q(t)$. So long as the galois group generated over $\Bbb Q(t)$ by $E_{\text{tor}}$ is all of $G$, ``almost all" specializations $t_0$ of $t$ in $\Bbb Q$ give rise to elliptic curves $E_{t_0}$ which are Serre curves, and if we consider those $t_0$ of height bounded by some $B$, we give bounds for the number of $E_{t_0}$ which are not Serre curves in terms of $B$.