We study the complexity of reachability problems on various classes of grid graphs. Reachability on certain classes of grid graphs gives natural examples of problems that are hard for NC/sup 1/ under AC/sup 0/ reductions but are not known to be hard far L; they thus give insight into the structure of L. In addition to explicating the structure of L, another of our goals is to expand the class of digraphs for which connectivity can be solved in logspace, by building on the work of Jakoby et al. (2001), who showed that reachability in series-parallel digraphs is solvable in L. We show that reachability for single-source multiple sink planar dags is solvable in L.