The result by Robertson and Seymour demonstrates an $O(n^3)$ algorithm for testing whether a fixed graph $G$ is a minor of $H$. I have two and a half questions on this topic:
1) It appears that there have been improvements to this algorithm since. What is the best-known algorithm at present?
2a) What do people conjecture to be the optimal bound?
Mohar's algorithm for embedding on a fixed surface and Kawarabayashi's algorithm for recognizing $k$-apex graphs decide membership of graphs characterizable by forbidden minors in linear time, motivating the last question:
2b) Is there any reason to suspect that we can do this in linear time?
Of course, if someone already came up with a linear-time algorithm, the last two questions are silly. :)