SC^2 algorithms for st-connectivity
There are two related complexity classes in $\text{NL}$ which are also in $\text{LogDCFL}$, which puts them in $\text{SC}^2$ (by Cook).  


The first is $\text{RUL}$, for "Reach-Unambiguous Log-space" which has reachability in mangroves (graphs where every pair of vertices has at most one directed path between them) as a complete problem.  This class has been discussed before.
The second is $\text{ReachFewL}$, which has reachability complete for graphs with at most a polynomial number of paths between any pair of vertices. 


Performing depth-first search on these graphs using a stack has a guarantee that it will take polynomial time, so these classes are in $\text{LogDCFL} \subseteq \text{SC}^2$.