  Conjecturally, a knot in the 3-sphere has only finitely many non-integer
non-characterizing slopes. We verify this conjecture for all knots with knot
Floer homology satisfying certain simplicity conditions. The class of knots
satisfying our notion of simplicity includes alternating knots, $L$-space knots
and the vast majority of knots with at most 12 crossings. For arbitrary knots
in the 3-sphere we show that almost all slopes $p/q$ with $|q|\geq 3$ are
characterizing. In addition, we show that all $L$-space knots and almost
$L$-space knots have infinitely many integer characterizing slopes.
