Not sure what "canonical" means in this context.  Half of algebraic geometry is called "canonical ring" or "fundamental class" or some other vacuous term.

Please consider Elkies [The Klein Quartic in Number Theory](http://library.msri.org/books/Book35/files/elkies.pdf) and in general the book **The Eightfold Way** is online.  In the translation of [Klein](http://library.msri.org/books/Book35/files/klein.pdf)'s original work we find this futuristic image:

[![enter image description here][1]][1]

Also in [German](https://eudml.org/journal/10142).  The translated version has improved images

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I have no idea what the above really means, but it sounds like we get really lucky.  These results are due to Riemann -- long before Serre, Grothendieck, Hartshorne or Shafarevich.  How could he have anything more complex in mind?

Since $x, y, z$ are the coordinates of an  equation $x^3 y + y^3 z + z^3 x = 0$, then $x, y, z$ can be thought of as parameterizing solutions to the set $$\{x^3 y + y^3 z + z^3 x = 0\}$$

So the coordinate ring might be the coordinate ring of $\mathbb{P}^N$ mod a single relation:
$$ \mathbb{C}[x,y,z]/(x^3 y + y^3 z + z^3 x ) $$
In fact, $x, y, z$ are **sections** of the line bundle $\mathcal{O}_X$ and if $i+j+k = n$ then maybe we can multiply them to get sections of the other sheafs:
$$ x^i y^j z^k \in H^0(X, n \mathcal{O}_X)$$
Looks like the coordinate ring is the direct sum of all of these possibilities excepting for the possibility that $x^3 y + y^3z + z^3 x = 0$.

The only ring mention in Klein's paper is $\mathbb{C}[x,y,z]/(x^3y + y^3z+z^3x)$ -- the one David Speyer wrote -- as well as $SL(2, \mathbb{F}_7)$ and $\mathbb{Z}(\frac{1 + \sqrt{-7}}{2})$.  Which one is the canonical ring?

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Resources include [Birational Geometry Old and New](http://www.ams.org/journals/bull/2009-46-01/S0273-0979-08-01233-0/home.html) (Antonella Grassi):

> **Example-Theorem 29**: If $C$ is a curve of genus $g \geq 2$ the divisor $3 K_C$ is *very ample*.  There is an embedding $\phi: C \to \mathbb{P}^N$ such that if $H$ is a hyperplane $H \cdot C = 3K_C$.  Then the canonical ring $R(C, K_C)$ can be reconstructed from the coordinate ring of $\mathbb{P}^N$.  In particular, $R$ is finitely generated.  In fact $3K_C$ determines a *pluricanonical embedding*. 

Grassi, in turn cites the standard texts:

* Hartshorne **Algebraic Geometry**
* Shafarevich **Basic Algebraic Geometry**


  [1]: https://i.sstatic.net/DRigf.png