Characterization of the $x$ such that $\sin(x)$ is rational? For $x \in [0,\pi/2]$, $\sin(x)$ ranges over $[0,1]$. 
So every rational number in $[0,1]$ is the sine of some $x \in [0,\pi/2]$.

Q. Is there any characterization of the $x$ for which $\sin(x)$ is rational?

I am not quite sure what shape such a characterization might take.
Something like: If $x$ satisfies such-and-such conditions, then $\sin(x)$ is rational. Perhaps there is at least a partial characterization?

First note that:
$x\to \sin(x)$ is locally a bijection on the interval you have given , so it's rational for an infinite number of arguments, in particular whenever it's evaluated on $\arcsin(r)$ for rational $r$
Another way of solving your problem is assuming that $\sin(x)$ and $\cos(x)$ are together rationals and here we will come up with the Pythagorean triples to solve the problem,if $$
(x,y)=\left( \frac{n^2-m^2}{n^2+m^2}, \frac{2mn}{n^2+m^2} \right).
$$
then $x^2+y^2=1$, so it's on the unit circle meaning that it's a value for some$(\sin(t),\cos(t))$
Finally, According to Niven's theorem in analysis which says that $\sin(r\pi)=q$ is rational if and only if $q=0,1,-1,\mp\frac{1}{2}$
