Dear Barinder,

Are you familiar with Fumiyuki Momose's "Isogenies of prime degrees over number fields?" If not, you may find it here on [NUMDAM][1] In it he performs an analysis of the isogeny character and finds that if $k$ is a quadratic field which is not a class number one imaginary quadratic field there are only finitely many $p$ for which $X_0(p)$ has noncuspidal rational points.

Furthermore if $k$ is any number field, a noncuspidal point of $X_0(p)(k)$ must be one of 3 types($\theta_p$ is the $p$-th cyclotomic character and $\lambda$ is the isogeny character of the point so $\lambda^{12}$ is independent of the representative elliptic curve or isogeny defining the point):

Type 1: $\lambda^{12}$ or $(\lambda\theta_p^{-1})^{12}$ is unramified 

Type 2: $\lambda^{12} = \theta_p^6$ and $p\equiv 3 \bmod 4$

Type 3: $k\supset H_L$, the hilbert class field of an imaginary quadratic field $L$ such that $p$ splits in $L$ and there are some further congruence conditions.


  [1]: http://www.numdam.org/numdam-bin/item?id=CM_1995__97_3_329_0