If $X$ is already a projective scheme, take $X' = X$ and $g = \mathrm{id}$. More generally, if $X$ is a proper scheme that is not projective (e.g., a complete non-projective toric variety), Chow's lemma produces a projective $X'$ with a birational map $X' \to X$.

Let $X$ be a proper scheme over $k$ and $Z \subset X$ a closed subscheme. The blowup $\mathrm{Bl}_Z(X) \to X$ is a proper birational morphism. If $\mathrm{Bl}_Z(X)$ is projective (which happens when $X$ is projective or when $Z$ is suitably chosen), this gives an explicit Chow cover. Hironaka's resolution of singularities provides Chow covers in characteristic zero.

Consider a proper smooth non-projective threefold $X$ over $\mathbb{C}$ (as in Hironaka's example). Chow's lemma guarantees a projective threefold $X'$ with a birational morphism $X' \to X$. Concretely, $X'$ can be obtained by blowing up appropriate subvarieties in a projective model.

Let $\mathcal{X}$ be a proper DM stack and $X$ its coarse moduli space (a proper algebraic space by Keel-Mori). Chow's lemma produces a projective $X' \to X$. Composing with $\pi: \mathcal{X} \to X$ gives a diagram $X' \to X \leftarrow \mathcal{X}$, relating the stack to a projective scheme.

An abelian variety $A$ over $k$ is proper. By the theory of ample line bundles on abelian varieties, $A$ is in fact projective, so Chow's lemma is trivially satisfied with $X' = A$. However, for quotients $A/G$ that are algebraic spaces (not schemes), Chow's lemma provides a nontrivial projective cover.

For a proper algebraic surface $X$ over a perfect field $k$, Chow's lemma combined with resolution of singularities for surfaces gives: there exists a smooth projective surface $X'$ and a birational morphism $X' \to X$. Moreover, $X'$ can be chosen to be obtained from $X$ by a sequence of blowups at closed points.

The coarse moduli space $\overline{M}_g$ is a proper algebraic space. Chow's lemma guarantees a projective $X' \to \overline{M}_g$. In fact, $\overline{M}_g$ is itself projective (by work of Knudsen, Mumford, and Cornalba-Harris), so Chow's lemma is superseded by this stronger result.

Chow's lemma is often used in proofs by reducing to the projective case. For instance, to prove the proper base change theorem for algebraic spaces, one uses Chow's lemma to reduce to the projective case (where the theorem is known for schemes) and then descends the result.

A K3 surface $X$ over $\mathbb{C}$ is proper and smooth. By a theorem of Kodaira, every K3 surface is Kähler, and the algebraic ones are projective. For an algebraic K3 surface (which is a scheme), Chow's lemma is automatic. Non-algebraic K3 surfaces exist in the analytic category but not in the algebraic one.

Over $\mathbb{C}$, for a proper algebraic space $X$, Chow's lemma combined with Serre's GAGA theorem gives: the categories of coherent sheaves on $X$ and on $X^{\mathrm{an}}$ are equivalent. The proof uses $X' \to X$ with $X'$ projective (where classical GAGA applies) and then descends.

For a proper morphism $f: X \to Y$ of noetherian algebraic spaces, there exists a projective $Y$-scheme $X'$ with a modification $X' \to X$ over $Y$. This relative version is essential for inductive arguments in the theory of algebraic spaces.

Let $G$ be a finite group acting on a proper algebraic space $X$. Then one can find an equivariant Chow cover: a projective scheme $X'$ with $G$-action and an equivariant modification $X' \to X$. This is proven by applying Chow's lemma to the stack $[X/G]$ and using the Keel-Mori correspondence.