Lindemann's prove of the transcendence of $\pi$ has settled the question, whether an arbitrary angle can be trisected, using straightedge and compass alone, to the negative.
In the following, trisectable always means with straightedge and compass alone.

I could however find nothing but some vague statements about angles, that are trisectable; a typical such statement is, that up to a few exceptions, it is impossible to trisect angles.

Question: I would therefore like to know, whether the angles, that are trisectable have been fully characterized and, if yes, who has done that first.