According the the introduction to Mazur's *Rational Isogenies of Prime Degree* the following question was open in 1978:



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It seems likely that this should have been resolved in the past 32 years. Does anyone know of a reference for this?



For which composite $N$ does $X_0(N)$ possess a non-cuspidal rational point?
Let me make David Brown's answer more explicit.


Theorem ([Kenku, 1979](http://alpha.math.uga.edu/%7Epete/Kenku79.pdf)): $X\_0(39)(\mathbb{Q})$ consists entirely of the ($4$) cuspidal points.


Theorem ([Kenku, 1980](http://alpha.math.uga.edu/%7Epete/Kenku80.pdf)): $X\_0(65)(\mathbb{Q})$ and $X\_0(91)(\mathbb{Q})$ each consist entirely of the ($4$) cuspidal points.


Theorem ([Kenku, 1980](http://alpha.math.uga.edu/%7Epete/Kenku80b.pdf), see also [Kenku, 1980](http://alpha.math.uga.edu/%7Epete/Kenku81.pdf)): $X\_0(169)(\mathbb{Q})$ consists entirely of $2$ rational cuspidal points.


Theorem ([Kenku, 1981](http://alpha.math.uga.edu/%7Epete/Kenku81b.pdf)): $X\_0(125)(\mathbb{Q})$ consists entirely of $2$ rational cuspidal points.


