The quickest description is to say that the constructible angles on the sphere are exactly the same as the constructible angles in the plane.
There are a few oddities. It is not possible to trisect a segment on the sphere, as this would involve trisecting an angle; think of the segment as part of the equator, and the north pole as part of a triangle with that.
It is possible to square the circle on the unit sphere. Not impressive, really. A hemisphere is both a square and a circle. There are a countable set of pairs of circular caps and symmetric quadrilaterals of the same area, where both are constructible.
Anyway, i wrote my article about the hyperbolic plane, not the unit sphere, but this material matches up.
April 2016, me to E. Rykken
Dear Prof. Rykken,
I have just read your article with Myles Dworkin on constructions on the sphere, and I have some suggestions. A little
late in the day, of course.
In preparing my Intelligencer article on constructions in the hyperbolic plane, I did not actually write out the analogous
statements on the unit sphere. However, it seemed to me very likely
that the angles that can be constructed on the sphere are exactly the
same as the angles that can be constructed in the plane. As to your
final paragraph, I would expect there to be a countable set of pairs
of circles and "squares" on the unit sphere, both constructable, with
equal area. However, i would expect these both need be constructed at
the same time, that there is no construction to begin with one and
reach the other.
Anyway, here are a copy of my old article and a page with Marvin's later article that incorporates some of it.