What is known about exceptional slopes of hyperbolic knots? For $K$ a hyperbolic knot in $S^3$, a rational number $p/q$ is an exceptional slope if the manifold $M_{p/q}$ obtained from $(p,q)$-Dehn surgery on $K$ does not admit a hyperbolic structure.
Thurston showed that there are finitely many exceptional slopes for any knot complement, and it's now known that there are at most $10$. I'm interested in understanding what sorts of numbers $p/q$ can occur. Is anything known in general?
For example, when $K$ is the figure-eight knot, the exceptional slopes are [T, Theorem 4.7] $1/0, 0/1, 1/1, 2/1, 3/1$, and $4/1$, along with their negatives. All of these except $(1,0)$ have $1$ in the denominator. Does this pattern hold more generally?
The exception to this pattern is $(1,0)$ which is always an exceptional slope because it gives $S^3$  for any $K$, unless I have the conventions backwards and $(0,1)$ does. (Usually we consider $1/0 = \infty$ to correspond to no surgery for this reason, but that doesn't seem to be Thurston's convention.)
[T] W. Thurston, Geometry and topology of three-manifolds, http://library.msri.org/books/gt3m/
 A: You will perhaps be interested in the paper "Non-integral toroidal Dehn surgeries" by Gordon and Luecke.  They prove:

Theorem 1.1. Let $K$ be a hyperbolic knot in $S^3$ that admits a non-integral surgery containing an incompressible torus. Then $K$ is one of the Eudave-Muñoz knots $k(l, m, n, p)$, and the surgery is the corresponding half-integral surgery.

Almost immediately before the statement they remark that the Eudave-Muñoz knots

... are the only known examples of non-trivial, non-integral, non-hyperbolic surgeries on hyperbolic knots.

A: For an overview, see the survey of Cameron Gordon
 and older survey.

Any integer can occur as the slope of a Seifert fibered Dehn filling on a knot.
Conjecture 3.3 of Gordon’s survey states that Seifert fillings must have integral slope. Thus one is left to consider reducible and toroidal fillings. Reducible fillings have integer slope. As indicated in Sam Nead’s answer, toroidal slopes are either integeral or half integral and one of Eudave-Muñoz’s examples. Hence the conjecture would give a complete characterization of exceptional Dehn filling slopes of hyperbolic knots. I’m not sure how much progress has been made on this conjecture in general. Exceptional surgeries have been classified for arborescent knots. Originally Gordon had conjectured that any Seifert surgery on a knot came from Dean’s construction, but other examples were found by Mattman-Katura-Kimihiko
. Other specific classes of Seifert spaces which can arise as Dehn fillings and classes of knots have been considered, as well as collections of exceptional slopes that can occur. But that seems to be the state of the art that is known about possible exceptional slopes.
