@aginensky: I think the OP was using the phrase "rational singularity" to mean "rational point which is a singular point" rather than the usual meaning from higher-dimensional geometry.

@aglearner: the quadric 3-fold is not toric.

@DavidRoberts: I agree with the second comment. The distinction made in your first comment seemed to be between "curve" and "Mordell--Weil group" or "group of rational points" (over some given field). That is what I was addressing.

@DavidRoberts: the distinction is important because a curve (over $\mathbf Q$ say) contains more information than just its group of rational points. For example, there are plenty of elliptic curves over $\mathbf Q$ whose group of rational points is trivial.

@PiotrAchinger: ironically my anecdote had to be withdrawn, or at least substantially revised.