We focus on various aspects of singular complex plane curves, mostly in the context
of their homological properties and the associated combinatorial structures. We
formulate some challenging open problems that can point to new directions in
research, for example, by introducing weak Ziegler pairs of curve arrangements.
Moreover, we construct new examples of different Ziegler pairs, in both the classical
and the weak sense, and present new geometric approaches to construction problems
of singular plane curves.
