What is Known

On page 5 of Wiles's paper, he discusses the history of the following theorem.

Theorem 2.1 (Gross, Kolyvagin, Zagier, et al.)   Suppose that

$$L(E,s) = c(s-1)^r +$$   higher order terms$$$$

with $r\leq 1$. Then the Birch and Swinnerton-Dyer conjecture is true for $E$, that is, $E(\mathbb{Q})\approx \mathbb{Z}^r\oplus E(\mathbb{Q})_{\tor}.$

I suspect that most elliptic curves satisfy the hypothesis of the above theorem, i.e., they have rank 0 or $1$. For example, almost 96% of the ``first $78198$'' elliptic curves have $r\leq 1$. I suspect that the curves with $r>1$ have ``density'' 0 amongst all elliptic curves. This doesn't mean that we are done. In practice it is often the curves with $r>1$ that are interesting and useful, and experts can still be observed saying ``almost nothing is known about the Birch and Swinnerton-Dyer conjecture''.