1. Which knots $K\subseteq S^3$ are such that there is a hyperbolic cone-manifold structure on $S^3$ that has (exactly) $K$ as the singular locus?
2. What if in Question 1 we restrict the cone angles to be $\leq \pi$?
3. Is it true that, if $M$ is a cone-manifold as in question 2, its volume is at most the simplicial volume of $S^3-K$ times $v\_3$?


I would also appreciate partial answers: it is known of some knots that aren't singularities of hyperbolic cone-manifold structures? Is it known of some non-hyperbolic knots that are? What is known about their volume?



Which knots are singularities of a hyperbolic cone-manifold structures on $S^3$?
The results proved in


S. Kojima, "Deformations of hyperbolic 3-cone-manifolds",
J. Differential Geom. 49 (1998), no. 3, 469-516 


provide complete answers to questions 1 and 3.


The main theorem of the cited paper ensures that any compact hyperbolic cone metric with cone angles at most π can be continuously deformed to a complete hyperbolic metric on the complement of the singularity. Furthermore this deformation can be done by an angle decreasing deformation. 


This provides a partial answer to your question (2) and a complete answer to (3): If $K$ is the singular locus of a cone structure with angles $\leq \pi$, then $K$ is necessarily hyperbolic. Moreover, thanks e.g. to a suitable version of the Schlafli formula, the hyperbolic volume of the cone structure is stricly smaller than the hyperbolic volume of the cusped manifold $S^3\setminus K$, which in turn is equal to the simplicial volume of $S^3\setminus K$ (up to the constant $v\_3$, which I guess you forgot to mention when asking question 3).


Moreover, Theorem 1.2.1 of Kojima's paper implies that, if $S^3$ supports a cone structure with singularity $K$, then $K$ must be hyperbolic, regardless of the cone angle of the original cone structure along $K$. 


