Section of universal curve

Let me assume $g>2$, so that there is a curve $C$ over $K_g$ of genus $g$ (the generic curve) -- the genus condition assures that the generic curve has no automorphisms. It is known but not obvious that $C$ has no $K_g$-rational points (this is due to Hain-Matsumoto in characteristic zero and Watanabe in positive characterstic, if I remember correctly). If you choose any closed point of $C$ with residue field $L$, the base change to $L$ will have a rational point.

There is a reasonably canonical choice of $L$. Namely, let $W_g$ be the moduli space of curves with a Weierstrass point. Then $W_g$ is connected and $W_g\to M_g$ is not Galois, but it is generically etale with monodromy group is $S_{g(g^2-1)}$ by this paper of Eisenbud-Harris. Thus letting $L_g$ be the function field of $W_g$, the generic curve obtains a rational point over $L_g$ (namely, the generic Weierstrass point). $L_g/K_g$ is not Galois, but its Galois closure has Galois group $S_{g(g^2-1)}$.

Of course this works in genus $<2$ as well, as long as you consider the generic point of $\mathcal{M}_g$ to be a stacky point rather than Spec of a field.

This got too long to be a comment, but...

Isn't $\mathcal{M}_g$ only a stack? Isn't the function field of a stack just the function field of its coarse moduli scheme (if it exists?)

For example, unlike in the case of schemes, I don't think there's a canonical map $Spec(K_g)\rightarrow\mathcal{M}_g$. Any such map is by definition given by an object of $\mathcal{M}_g$ over $K_g$, which is precisely the pullback of $\mathcal{M}_{g,1}\rightarrow\mathcal{M}_g$ to $K_g$.

For example, if you replace $\mathcal{M}_g$ with $\mathcal{M}_{1,1}$, then there are infinitely many elliptic curves over $\mathbb{Q}(j)$ with $j$-invariant $j$, corresponding to infinitely many maps $Spec\;\mathbb{Q}(j)\rightarrow\mathcal{M}_{1,1}$, each of which can be thought of as a "generic point", and none of them seem to be preferable over any other.

See for example Noam Elkies' answer to: Questions about the "universal elliptic curve" over the affine $j$-line punctured at 0 and 1728