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 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.