The complexity of determining if a fixed graph is a minor of another 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. :)
There is a preprint by Ken-ichi Kawarabayashi, Yusuke Kobayashi, and Bruce Reed that claims a quadratic time algorithm: "The disjoint paths problem in quadratic time". It is formatted as a conference submission rather than a journal paper so I'm not sure it's possible to verify the details, though (I haven't really tried, myself).

A very recent survey by Kawarabayashi cites this as the best known result for the closely related disjoint paths problem: Ken-ichi Kawarabayashi (2011), "The Disjoint Paths Problem: Algorithm and Structure", WALCOM: Algorithms and Computation, LNCS 6552, pp. 2–7, doi:10.1007/978-3-642-19094-0_2.

I don't know whether this means that the $O(n\log n)$ claim in Kothari's comment is vapor or whether it means that it's still at an earlier stage of being written up.