Quasi-\'etale covers of Du Val del Pezzo surfaces and Zariski dense exceptional sets in Manin's conjecture

We construct first examples of singular del Pezzo surfaces with Zariski dense exceptional sets in Manin's conjecture, varying in degrees $1, 2$ and $3$. To systematically study these examples, we classify all quasi-\'etale covers of Du Val del Pezzo surfaces up to singularity types and study their equivariant geometry.