What is the notation for saying $S$ is a subring of $R$?

This is correct. There is a notation for ideal, yet no notation for subring, as far as I know. One just writes: "Let $S \subset R$ be a subring."

Anyway, also the notation for subgroup, ideal and alike is not used all that much. "Let $H\subset G$ be a subgroup." Is what I see all the time.

I have used and seen $S\leq R$ to say that $S$ is a substructure of $R$.

As you asked for the notation, I read this as how subrings are identified in the literature so I looked up the definitions for subrings in a few well known books. I checked the books on Algebra by Artin, Dummit & Foote, Hungerford, Jacobson, Lang, van der Waerden, where notation for a subring seems to be nonexistent, i.e., "is a subring" is as good as it gets, at least in their definition of what a subring is.

Fraleigh in A First Course in Algebra, p.173 holds off explicitly defining any notation, but states:

... In fact, let us say here once and for all that if we have a set, together with a certain type of algebraic structure (group, ring, field, integral domain, vector space, and so on), then any subset of this set, together with the natural induced algebraic structure that yields an algebraic structure of the same type, is a substructure. If $K$ and $L$ are both structures, we shall let $K\le L$ denote that $K$ is a substructure of $L$ and $K<L$ denote that $K\leq L$ but $K\neq L$.

Then in Rotman's Modern Algebra, he defines a subring then gives the following on p.119:

Notation. In contrast to the usage $H \leq G$ for a subgroup, the tradition in ring theory is to write $S \subseteq R$ for a subring. We shall also write $S \subsetneq R$ to denote a proper subring; that is, $S \subseteq R$ and $S \neq R$.

So it seems fair to say $S\subseteq R$, and $S\leq R$ can be safely used in place of any explicit notation reserved for subrings.