Let $\mathcal{B}(H)$ denote the C*-algebra of all bounded operators on a *separable* infinite dimensional complex Hilbert space $H$. It is a well-known fact that $\mathcal{B}_0(H)$, the ideal of compact operators, is the unique, nontrivial, proper *closed* ideal in $\mathcal{B}(H)$. Furthermore, any proper ideal in $\mathcal{B}(H)$ must be contained in $\mathcal{B}_0(H)$.

Is there any reasonable characterization of all (not necessarily closed) ideals in $\mathcal{B}(H)$ which contain the ideal of finite rank operators (possibly under some definability constraints, like being Borel in the strong operator topology, etc)?

Also, not being an operator theorist, are there important ideals of this type other than $\mathcal{B}_0(H)$ and the Schatten $p$-ideals (including the trace-class and Hilbert-Schmidt operators)?