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One result along those lines is that that any algebraic space which has a quasi-finite morphism to a scheme is itself a scheme.

More precisely, if $f:X\to Y$ is a separated, locally quasi-finite, locally finite type morphism from an algebraic space to a scheme, then by the Stein factorization theorem, $f$ is quasi-affine, so $X$ must be a scheme. If you want more details, this is Corollary 17.8 in my notes from Martin Olsson's stacks course.

(Martin has since written a book covering the material from that course. I don't have a copy on hand to check, but this should be in section 7.2 on Stein factorization.)

One result along those lines is that that any algebraic space which has a quasi-finite morphism to a scheme is itself a scheme.

More precisely, if $f\colon X\to Y$ is a separated, finite type, quasi-finite morphism of algebraic spaces, then the Stein factorization $X\to \mathrm{Spec}_Y(f_*\mathcal{O}_X)\to Y$ is an open immersion followed by an affine morphism, so $f$ is quasi-affine. In particular, if $Y$ is a scheme, so is $X$.

References

This is Proposition 3.1 of Quot Functors for Deligne-Mumford Stacks or Théorème A.2 of Champs algébriques.

I learned this from Martin Olsson; it's Corollary 17.8 in my notes from his stacks course. It's Theorem 7.2.10 in the book he's since written covering the material from that course. Jason Starr points out that there's a mistake in Lemma 6.2.9 which propagates to this theorem (I haven't checked this), and suggested the two alternative references above.

One result along those lines is that that any algebraic space which has a quasi-finite morphism to a scheme is itself a scheme.

More precisely, if $f:X\to Y$ is a separated, locally quasi-finite, locally finite type morphism from an algebraic space to a scheme, then by the Stein factorization theorem, $f$ is quasi-affine, so $X$ must be a scheme. If you want more details, this is Corollary 17.8 in my notes from Martin Olsson's stacks course.

One result along those lines is that that any algebraic space which has a quasi-finite morphism to a scheme is itself a scheme.

More precisely, if $f:X\to Y$ is a separated, locally quasi-finite, locally finite type morphism from an algebraic space to a scheme, then by the Stein factorization theorem, $f$ is quasi-affine, so $X$ must be a scheme. If you want more details, this is Corollary 17.8 in my notes from Martin Olsson's stacks course.

(Martin has since written a book covering the material from that course. I don't have a copy on hand to check, but this should be in section 7.2 on Stein factorization.)

One result along those lines is that that any algebraic space which has a quasi-finite morphism to a scheme is itself a scheme. I'll go find a reference now.

One result along those lines is that that any algebraic space which has a quasi-finite morphism to a scheme is itself a scheme.

More precisely, if $f:X\to Y$ is a separated, locally quasi-finite, locally finite type morphism from an algebraic space to a scheme, then by the Stein factorization theorem, $f$ is quasi-affine, so $X$ must be a scheme. If you want more details, this is Corollary 17.8 in my notes from Martin Olsson's stacks course.