Supersingular elliptic curves over $\mathbb{Q}$ what are the examples of elliptic curves defined over  $\mathbb{Q}$ with supersingular reduction at a prime $p$ and having a $p$-isogeny over $\mathbb{Q}$ ?
 A: In fact, this cannot happen: an elliptic curve over $\mathbb{Q}_p$ is supersingular if and only if its associated mod $p$ Galois representation is irreducible, but if it is irreducible as a representation of $\mathbb{F}_p[G_{\mathbb{Q}_p}]$ then it is certainly irreducible as a representation of $\mathbb{F}_p[G_{\mathbb{Q}}]$ and thus it has no $p$-isogenies.
This argument doesn't work over larger fields: if $K$ is an extension of $\mathbb{Q}_p$ it's no longer true that $E/K$ is supersingular if and only if $E[p]$ is irreducible as a $\mathbb{F}_p[G_{K}]$-module, cf. this MSE question. In particular, one can have an elliptic curve $E / \mathbb{Q}$ with a $p$-isogeny and bad reduction at $p$, and a number field $F / \mathbb{Q}$ such that $E$ has good supersingular reduction at all primes of $F$ above $p$; my previous suggestion to look for points on $X_0(p)$ gives lots of examples of this.
A: I felt that there couldn’t be such an isogeny, but when I saw @DavidLoeffler’s answer, I realized that I had an argument, too. The formal group of the elliptic curve would be of height two, defined over $\mathbb Z_p$, but such things don’t have $p$-isogenies: the quotient formal group is definable only over a suitably ramified extension.
