The curve $y^2+y=x^3-x$ has order 5 mod 2 and order 7 mod 3. Therefore the torsion injects into both $Z/5Z$ and $Z/7Z$ and hence is trivial. This is the first example in Husemoller's book on Elliptic Curves.
As Zev notes in his comment, this is the tip of a very large iceberg.
