Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$.
Say that $S$ is an *integer-distance set* if every pair of points in $S$ is separated
by an integer Euclidean distance.

What are examples of maximal integer distance sets? (Maximal: no point can be added while retaining the integer-distance property between all pairs.)

Of course the lattice points along any one line parallel to a coordinate axis in $\mathbb{R}^d$ constitute a countably infinite integer-distance set. What is an example of an infinite integer-distance set of noncollinear points?

I know that Euler established that every circle contains a dense rational-distance set.
Scaling any one circle by a large common denominator provides a rich, but finite integer-distance set. For example, these five points on a circle are all separated by integer
distances:
$$
(1221025, 0), (781456, 586092),
(439569, 586092),
(270400, 507000),
(180625, 433500)
$$

I'm sure this is all known... Thanks for enlightening me!

(This is tangentially related to my earlier question, "Rational points on a sphere in $\mathbb{R}^d$.")

**Update1.**It turns out that determining the integer-distance sets is fundamentally open. What is known is nicely summarized by Robert Israel and "Daniel m3." In particular, via the Kreisel & Kurz paper, it is unknown (or was unknown in 2008) whether or not there exists an 8-point integer-distance set in $\mathbb{R}^2$, with no three of the points collinear and no four cocircular.

**Update2.**
Also open is a related problem identified by Nathan Dean:
How many non-cocircular integer-distance points can be found on a *parabola*,
a scaling of $y = x^2$?
Nathan proved there are infinitely many sets of three such points;
Garikai Cambell proved there are infinitely many sets of four such points.
But the existence of five such points seems open.
I just learned the parabola problem from this MSE question.

**Update3.**
I ran across this just-published paper, which explores the in-some-sense obverse of the
question I asked: What are the largest point sets in $\mathbb{R}^d$ that
avoid points an integral distance apart.

Kurz, Sascha, and Valery Mishkin. "Open Sets Avoiding Integral Distances."

Discrete & Computational Geometry(2013): 1-25. (Springer link)