Efficient logspace algorithms
I guess Single-source Single-sink Planar DAG (SSPD) reachability has logspace algorithm with a modest running time ($O(n^2)$?). I am not so sure about
Single-source Multiple-sink Planar DAG Reachability (SMPD) algorithm.

Ref: Eric Allender, David A. Mix Barrington, Tanmoy Chakraborty, Samir Datta, Sambuddha Roy: Planar and Grid Graph Reachability Problems. Theory Comput. Syst. 45(4): 675-723 (2009)

Also, a new logspace algorithm for planarity testing and embedding runs in modestly polynomial time (modulo undirected reachability, of course)

Ref: Samir Datta, Gautam Prakriya: Planarity Testing Revisited CoRR abs/1101.2637: (2011)

Finally, here is a simple toy problem which has a logspace algo with a modest
running time (modulo undirected reachability) viz. Outerplanar Isomorphism.