What are examples of $\mathsf{NL}$-complete problems?
Here are three more examples, taken from an assignment by Trevisan:


Given an NFA and a word, determining whether the NFA accepts the word.
Reachability in DAGs.
Determining whether a digraph is strongly connected.


Planken shows that the Simple Temporal Problem with bounded weights is NL-complete (see the link for definitions).

There are likely many others. Schaefer's dichotomy theorem, in particular, gives many examples.