nt.number theory - Are most cubic plane curves over the rationals elliptic?

Your question (as explained in the second paragraph) is not vague at all! In fact, it appears for instance after Conjecture 2.2 in http://www-math.mit.edu/~poonen/papers/random.pdf , which is Random diophantine equations, B. Poonen and J. F. Voloch, pp. 175–184 in: Arithmetic of higher-dimensional algebraic varieties, B. Poonen and Yu. Tschinkel (eds.), Progress in Math. 226 (2004), Birkhäuser.

The answer is not known, and the experts I've spoken to do not even have a convincing heuristic predicting an answer. Swinnerton-Dyer told me that he had a hunch that the answer was 0, and this is my hunch too, but we have little to back this up.

It is not even clear that the limit exists. One can prove, however, that the density (in your precise sense) of plane cubic curves that have points over $mathbb{Q}_p$ for all $p le infty$ is a number strictly between $0$ and $1$ (Theorem 3.6 in the Poonen-Voloch paper), so the lim sup of the fraction of plane cubic curves with a rational point is at most this; in particular, it's not 1.

One could try to estimate the size of the Tate-Shafarevich group of a "random" elliptic curve, to get an idea of how often local solvability implies global solvability, but even if one does this it is not clear that this is counting curves in the same way.