Testing $H$ a topological minor of $G$ --- algorithms that scale with $H$?
The general problem when $H$ is not fixed is NP-hard: Take $H$ to be a cycle of length $|V(G)|$ and it amounts to testing whether $G$ is hamiltonian.

I think that the state of the art on this is the FPT algorithm of 
Grohe, Kawarabayashi, Marx and Wollan. The dependency on $G$ is cubic, but the one on $H$ seems impractical, since it relies quite heavily on the Robertson-Seymour theory.