what is the maximum number of rational points of a curve of genus 2 over the rationals Conjecturally, there exists an integer $n$ such that the  number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.)
We are very far from proving the existence of such an integer, let alone find an explicit value which works.
My question is:
What is the best known lower bound for $n$?
One way to obtain a lower bound $m$ for $n$ is to prove the existence of a curve of genus $2$ over $\mathbf{Q}$ with at least $m$ rational points. 
 A: I believe it is 642. See http://www.mathe2.uni-bayreuth.de/stoll/recordcurve.html
A: Here is some more information.

*

*The curve that establishes the current record is obtained from a K3
surface $S$ that was found by Noam Elkies. $S$ is a double cover of
${\mathbb P}^2$ ramified above a smooth sextic $B$ that has lots of
tritangent lines and also higher degree curves meeting it with even
intersection multiplicity in all points.
Pulling back any rational line $L$ in ${\mathbb P}^2$ to $S$ that is not
tangent to $B$ gives a genus 2 curve $C$ on which all the tritangents induce
pairs of rational points, and the same is true for the higher degree
curves when they intersect $L$ in rational points. Of course, there can
be additional rational points on $C$ that do not arise in this way.
I found the record curve by a systematic search through rational lines
of relatively small height. The previous record was 588 points, due to
Keller and Kulesz. Their curve is of a special form and has 12 automorphisms
defined over $\mathbb Q$; the 588 points come in 49 orbits. By contrast,
the new record curve has minimal automorphism group (only the hyperelliptic
involution).


*The result of Caporaso, Harris and Mazur needs the weak Lang conjecture
for varieties of arbitrarily large dimension. (I have seen Bombieri protest
against the name `Bombieri-Lang Conjecture', saying that he only ever
made the conjecture for surfaces.) We know essentially nothing in this
direction beyond what is covered by Faltings' result on subvarieties of
abelian varieties (or can be deduced from that).


*What is perhaps more convincing is the conjecture that there should be
a bound in terms of the genus $g$ and the rank $r$ for the number of
(geometric) points
on a genus $g$ curve $C$ mapping into a rank $r$ subgroup of its Jacobian
(under some embedding given by a base-point on $C$). This follows from
the Zilber-Pink conjecture for families of abelian varieties. This of
course implies a bound for the number of rational points on a genus 2 curve
in terms of the Mordell-Weil rank of its Jacobian. If these ranks are
bounded, then one would expect a bound on the number of rational points.
But this is another open question. Perhaps the data in
this paper might be interesting.
EDIT: This statement ("Uniform Mordell-Lang") has by now
been proved in work of Lars Kühne,
building on work
of Dimitrov, Gao and Habegger. Another proof was given
by Xinyi Yuan.


*Such bounds do exist (and are even explicit) when the rank $r$ is sufficiently small compared to $g$; concretely for $r \le g-3$ (which
unfortunately does not tell us anything about the case $g = 2$). See this and this paper. For hyperellptic curves, the bound can
be taken to be $33(g-1) + 8rg \pm 1$  ($+$ when $r = 0$, $-$ otherwise).

A: The way the question is formulated, the answer is unbounded,
since rational singular points can be unbounded.
For natural $n$, define $j(n)=\prod_{i=1}^n(x-i)$.
Consider the curve $C_n : j(n)^2(x^5+13)=y^2$.
It is birationally equivalent to $x'^5+13=y'^2$ which is genus $2$,
so $C_n$ is genus $2$.
$C_n$ has the rational (singular) points $(1,0),(2,0),\ldots(n,0)$
which are unbounded, since $n$ is unbounded.
