Thank you all for your very kind answers! It was silly of mine to suggest we rate the examples, since it's unclear if any examples are better than others. To atone for my mistake, let me offer a summary of the proposed examples in the form of a table.

## stack : coarse moduli space

$\mathcal{M}_1$ : affine line $\mathbf{A}^1$;

line bundles of degree $0$ on a smooth curve : $\mathrm{Pic}^0$;

$[\mathrm{pt}/G]$ : $\mathrm{pt}$.

## Good moduli spaces

One thing that I learned from Alper, Good moduli spaces is that an object technically better than the coarse moduli space is a *good moduli space*, a replacement notion which commutes with arbitrary, not just flat, base change, and exists more generally. Explicitly, a good moduli space of an Artin stack $X$ is a morphism $f$ to an algebraic space such that 0) it's quasi-compact a) pushforward along $f$ is exact on quasi-coherent sheaves, and b) the pullback morphism $f_*\mathcal{O}_Y\rightarrow \mathcal{O}_X$ is an isomorphism.

For example, given a linear algebraic group $G$ acting an an affine scheme of ring A over a field, the morphism
$$
[\mathrm{Spec}(A)/G] \rightarrow \mathrm{Spec}(A^G)
$$
is a good moduli space (ibid., Example 8.3), hence over $\mathbf{Q}$,
$$
[\mathbf{A}^1/\mathrm{Spec}(k)]\rightarrow \mathrm{Spec}(k)
$$
is a good moduli space (ibid., Example 8.1 and Example 12.4 (1)).

And when is the old notion of a coarse moduli space a special case of the new notion of good moduli space, you ask? Well, if the stack in question is *tame* (ibid. Example 8.1), i.e., if it has inertia stack finite over it and if the morphism from it to the coarse moduli space is exact on quasicoherent sheaves. (The latter condition is automatic in characteristic $0$.)