The figure eight-knot has pseudo-Anosov monodromy with no singularity. I have read that the (-2,3,7)-pretzel knot has pseudo-Anosov monodromy with a single 18-prong singularity on the boundary of the fiber.
My question is : do we know a class of hyperbolic fibered knots in $S^3$ whose monodromy has (at most) a single singularity on the boundary of the fiber ?