According to several sources, it is conjectured (or at least believed) that the rational points of curves over the rationals of genus $g > 1$ are uniformly bounded by $g$. E.g. here p. 1.

Assuming the curve is irreducible, the singular points (which are also rational) can easily be made unbounded for fixed $g > 1$.

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 birationally equivalent to $x'^5+13=y'^2$ which is genus $2$.

$C_n$ has the rational (singular) points $(1,0),(2,0),\ldots(n,0)$ which are unbounded, since $n$ is unbounded.

The same applies for $2$ replaced by integer $d>1$.

For $d > 2$ the curve need not be hyperelliptic.

Q. What is the exact statement about uniform boundedness?

According to both sage and magma, $C_n$ is irreducible for small values of $n$.

arithmetic genusof that curve grows as the number of singular points. So if you want a boundedness result for singular curves, you should work with the arithmetic genus. $\endgroup$geometricgenus by "genus". $\endgroup$