The stack $\mathcal{M}_g$ ($g \geq 2$) has coarse moduli space $M_g$, a quasi-projective variety of dimension $3g - 3$. The map $\pi : \mathcal{M}_g \to M_g$ is an isomorphism on the open locus of curves with trivial automorphism group (which is dense for $g \geq 3$).

For $g = 2$: every genus-2 curve is hyperelliptic, so $\operatorname{Aut}(C) \supseteq \mathbb{Z}/2\mathbb{Z}$ for all $C$. The coarse space $M_2$ has dimension 3 and is rational (birationally equivalent to $\mathbb{A}^3$).

The Deligne--Mumford compactification $\overline{\mathcal{M}}_g$ has coarse moduli space $\overline{M}_g$, a projective variety. Mumford proved $\overline{M}_g$ is of general type for $g \geq 24$ (later improved by Harris--Mumford to $g \geq 23$, and by Farkas to $g = 22$).

The stack $\mathcal{M}_{1,1}$ of elliptic curves has coarse moduli space $M_{1,1} \cong \mathbb{A}^1_j$ (the $j$-line). The map $\pi : \mathcal{M}_{1,1} \to \mathbb{A}^1_j$ sends an elliptic curve $E$ to its $j$-invariant.

Over $\mathbb{C}$, $\pi$ is an isomorphism over $\mathbb{A}^1 \setminus \{0, 1728\}$ (where automorphism groups are $\mu_2$, the minimum). At $j = 1728$, the fiber has automorphisms $\mu_4$, and at $j = 0$, the fiber has automorphisms $\mu_6$.

There is no universal elliptic curve over $\mathbb{A}^1_j$. The obstruction is precisely that $\mathcal{M}_{1,1} \to \mathbb{A}^1_j$ is a nontrivial $\mu_2$-gerbe generically (every elliptic curve has the $[-1]$ involution).

For a finite group $G$, the classifying stack $\mathrm{B}G = [\operatorname{Spec} k / G]$ has coarse moduli space $\operatorname{Spec} k$ (a point). The map $\pi : \mathrm{B}G \to \operatorname{Spec} k$ collapses all the automorphism data. Over an algebraically closed field, $\mathrm{B}G$ has a unique isomorphism class (the trivial torsor), so the bijection on geometric points holds.

This is the simplest example where the coarse space is smooth but the stack is "non-trivially stacky." The coarse map $\pi_* : \operatorname{QCoh}(\mathrm{B}G) \to \operatorname{QCoh}(\operatorname{Spec} k)$ sends a $G$-representation $V$ to its invariants $V^G$.

Let $G$ be a finite group acting on a smooth variety $X$. The quotient stack $[X/G]$ is a smooth DM stack, and its coarse moduli space is the GIT quotient $X/G$. When $G$ acts with fixed points, $X/G$ acquires singularities even though $[X/G]$ is smooth.

For $G = \mu_n$ acting on $\mathbb{A}^2$ by $\zeta \cdot (x,y) = (\zeta x, \zeta^{-1} y)$: the stack $[\mathbb{A}^2/\mu_n]$ is smooth, but the coarse space is the $A_{n-1}$ singularity $\operatorname{Spec} k[u,v,w]/(uv - w^n)$.

This illustrates a key principle: stacky smoothness is weaker than scheme smoothness. The stack remembers the smooth local structure (it is locally the quotient of a smooth space), while the coarse space has a singularity at the fixed point.

The weighted projective stack $\mathcal{P}(a_0, \ldots, a_n)$ has coarse moduli space $\mathbb{P}(a_0, \ldots, a_n)$, the classical weighted projective variety. When the weights are not all equal, $\mathbb{P}(a_0, \ldots, a_n)$ has cyclic quotient singularities at the coordinate vertices.

For $\mathcal{P}(1, 1, 2)$: the coarse space $\mathbb{P}(1,1,2)$ is isomorphic to the quadric cone $V(xy - z^2) \subset \mathbb{P}^3$ (it has an $A_1$ singularity at the vertex $[0:0:1]$). The stack $\mathcal{P}(1,1,2)$ is smooth.

The stack $\mathcal{A}_g$ of principally polarized abelian varieties has coarse moduli space $A_g$. Over $\mathbb{C}$, $A_g = \mathrm{Sp}(2g, \mathbb{Z}) \backslash \mathfrak{H}_g$ (the Siegel modular variety).

For $g = 1$: $A_1 = \mathbb{A}^1_j$ (the $j$-line), since principally polarized abelian varieties of dimension 1 are elliptic curves.

For $g = 2$: $A_2$ is a 3-dimensional variety. The Torelli map $M_2 \to A_2$ is an isomorphism onto its image (since every principally polarized abelian surface that is a Jacobian is the Jacobian of a unique curve of genus 2).

For $g = 3$: $A_3$ has dimension 6, and the Torelli map $M_3 \to A_3$ is generically injective (but not surjective: the image has codimension 0 since $\dim M_3 = 6 = \dim A_3$, but not all ppav of dimension 3 are Jacobians -- this is the Schottky problem).

The root stack $\sqrt[r]{D/X}$ (for a smooth divisor $D \subset X$) has coarse moduli space $X$ itself. The map $\pi : \sqrt[r]{D/X} \to X$ is an isomorphism over $X \setminus D$ and is a $\mu_r$-gerbe along $D$.

For the pushforward: if $\mathcal{F}$ is a coherent sheaf on $\sqrt[r]{D/X}$, then $\pi_*\mathcal{F}$ is a coherent sheaf on $X$ with a filtration determined by the $\mu_r$-action on the fibers over $D$. This connects to the theory of parabolic sheaves.

The moduli stack of semistable Higgs bundles $\mathcal{H}iggs_n^{\mathrm{ss}}(C)$ has a coarse moduli space $\mathcal{M}_H$ (the Hitchin moduli space), which is a quasi-projective variety. For $n = 2$ on a genus-$g$ curve: $\dim \mathcal{M}_H = 2(4(g-1)) = 8g - 8 \quad (\text{as real manifold: } \dim_{\mathbb{R}} = 16g - 16).$ Actually for rank $n$ and genus $g$: $\dim \mathcal{M}_H = 2n^2(g-1)$.

The Hitchin map $h : \mathcal{M}_H \to \mathcal{B}$ (to the Hitchin base) makes $\mathcal{M}_H$ into a completely integrable system. The generic fiber is an abelian variety (the Prym of the spectral curve).

Let $\mathcal{X} \to X$ be a $\mu_n$-gerbe over a scheme $X$. Then the coarse moduli space of $\mathcal{X}$ is $X$ itself. The coarse map $\pi : \mathcal{X} \to X$ forgets the gerbe structure entirely.

For the pushforward of sheaves: $\pi_*\mathcal{O}_\mathcal{X} = \mathcal{O}_X$, but for a line bundle $\mathcal{L}$ of weight $j$ (under the $\mu_n$-action) on $\mathcal{X}$, $\pi_*\mathcal{L} = 0$ if $j \not\equiv 0 \pmod{n}$. This is because the $\mu_n$-invariant part of a weight-$j$ representation is zero for $j \neq 0$.

Let $E$ be an elliptic curve with involution $\iota : z \mapsto -z$. The stack $[E/\langle\iota\rangle]$ is a smooth DM stack. The coarse moduli space is $E/\langle\iota\rangle \cong \mathbb{P}^1$ (the Weierstrass model). The four fixed points of $\iota$ (the 2-torsion points) become orbifold points of order 2 in the stack, and ordinary double points are not introduced in $\mathbb{P}^1$ (which is already smooth), but the map $[E/\langle\iota\rangle] \to \mathbb{P}^1$ has nontrivial $\mu_2$-stabilizers at 4 points.

The pushforward $\pi_*\mathcal{O}_{[E/\langle\iota\rangle]} = \mathcal{O}_{\mathbb{P}^1}$, consistent with $H^0(E, \mathcal{O}_E)^{\langle\iota\rangle} = k$.

The Kontsevich stack $\overline{\mathcal{M}}_{0,n}(\mathbb{P}^r, d)$ of degree-$d$ rational stable maps to $\mathbb{P}^r$ is a proper DM stack. Its coarse moduli space $\overline{M}_{0,n}(\mathbb{P}^r, d)$ is a projective variety. For $r = 2$ and $d = 1$: $\overline{M}_{0,0}(\mathbb{P}^2, 1) \cong (\mathbb{P}^2)^*$ (the dual projective plane, parametrizing lines in $\mathbb{P}^2$).

The virtual fundamental class lives on the stack $\overline{\mathcal{M}}_{0,n}(\mathbb{P}^r, d)$, but Gromov--Witten invariants (intersection numbers) can be computed on the coarse space since the pushforward $\pi_*$ preserves degree for proper maps (at least in characteristic zero, where the orders of stabilizers are invertible).

The coarse moduli space is always an algebraic space but may not be a scheme. An example: let $\mathcal{X}$ be a smooth DM stack whose coarse space $M$ is an algebraic space that is not a scheme (such examples exist in dimension $\geq 3$, cf. Hironaka's example).

More commonly: $M_g$ is a scheme for $g \geq 2$ (in fact quasi-projective), $\overline{M}_g$ is projective, $A_g$ is quasi-projective. For these classical moduli problems, the coarse spaces are always schemes.