Singular plane curves: freeness and combinatorics

In this paper 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.