This question is wildly open still in characteristic zero.

We know some examples of curves that are s.c.i.’s, for example Rao’s classification of self-linked curves, Hartshorne and Polini’s examples of curves on ruled cubic surfaces.

In Hartshorne and Polini’s recent paper on codepth, they reformulated the conditions for a curve to be an s.c.i. in terms of the new notion of codepth, and gave several necessary criteria.

One interesting necessary condition for $C$ in $P^3$ to be an s.c.i. is that $C$ must be contained in a hypersurface $X$ such that $P^3-X$ is affine. Thus an example of a smooth curve $C$ in $P^3$ not contained in any such hypersurface would give a counter example to the s.c.i. conjecture.

It is believed that the rational quartic is not likely an s.c.i, however no complete proof has been published yet. The best so far are results on the lower bounds of the degrees for the two equations should they exist.

This question is wildly open still in characteristic zero.

We know some examples of curves that are s.c.i.’s, for example Rao’s classification of self-linked ACM curves, Hartshorne and Polini’s examples of curves on ruled cubic surfaces.

In Hartshorne and Polini’s recent paper on codepth, they reformulated the conditions for a curve to be an s.c.i. in terms of the new notion of codepth, and gave several necessary criteria.

One interesting necessary condition for $C$ in $P^3$ to be an s.c.i. is that $C$ must be contained in a hypersurface $X$ such that $X-C$ is affine. Thus an example of a smooth curve $C$ in $P^3$ not contained in any such hypersurface would give a counter example to the s.c.i. conjecture.

It is believed that the rational quartic is not likely an s.c.i, however no complete proof has been published yet. The best so far are results on the lower bounds of the degrees for the two equations should they exist.