How many points can you find on $y=x^2$, for $x \geq 0$, such that each pair of points has rational distance?
The answer to the "infinite family" question appears to be, no. Jozsef Solymosi and Frank de Zeeuw, On a question of Erdős and Ulam, Discrete Comput. Geom. 43 (2010), no. 2, 393–401, MR2579704 (2011e:52024), prove (according to the review by Liping Yuan) that no irreducible algebraic curve other than a line or a circle contains an infinite rational set.