This question is related to the previous discussion here.
Due to the result of Noga Alon et al., there is an $O((2k)^kn)$ algorithm for deciding whether a planar graph $G$ contains a fixed subgraph $H$ of size $k$, and the time complexity is reduced to $O(2^kn)$ if the graph $H$ is of bounded treewidth. Take $k = O(\log n)$ yields a polynomial time algorithm for the latter case, say the $k$-path problem mentioned by Ryan Williams in this paper.
There is an open problem in the result:
If we want to solve $k$-path problem in a planar graph with slightly larger $k$, say $k = O(\log^2 n)$, is there a polynomial time solution at this point? If so, what is the best time complexity at present?