In a recent work, we initiated the study of Borel equivalence relations defined on the Polish space ${\rm{SA}}(H)$ of self-adjoint operators on a Hilbert space $H$, focusing on the difference between bounded and unbounded operators. In this paper, we show how the difficulty of specifying the domains of self-adjoint operators is reflected in Borel complexity of associated equivalence relations. More precisely, we show that the equality of domains, regarded as an equivalence relation on ${\rm{SA}}(H)$, is continously bireducible with the orbit equivalence relation of the standard Borel group $\ell^{\infty}(\mathbb{N})$ on $\mathbb{R}^{\mathbb{N}}$. Moreover, we show that generic self-adjoint operators have purely singular continuous spectrum equal to $\mathbb{R}$.