Are there problems that are DSPACE(O(1)) complete?
YES, there are.

At the beginning of "Andreas Krebs, Klaus-Jörn Lange: Dense Completeness. Developments in Language Theory 2012: 178-189"[Section 6], it is stated that some REG languages with a non-solvable monoid are complete for $\mathrm{NC}^1$ under $\mathrm{DLOGTIME}$-uniform $\mathrm{AC}^0$ reductions. The section goes on to state that a REG language is either in $\mathrm{AC}^0$ or its syntactic monoids are non-aperiodic.

Complexity Zoo states that REG is contained in $\mathrm{NC}^1$.

So, some REG language is complete for the class itself. This is non-trivial.