Every isometry of a proper $CAT(-1)$ space is either elliptic, parabolic, or hyperbolic: elliptic means fixes a finite point; hyperbolic means fixes two infinite points connected by a translation axis, equivalently translation distance bounded away from zero; parabolic means translation distance limiting to zero but no fixed point. There are versions for proper Gromov hyperbolic spaces, and even for the nonproper case, if you are willing to "quasify" the statements of the cases, and if you are willing to let the trichotomy degenerate to a dichotomy.
Every isometry of Teichmuller space is either elliptic, parabolic, or hyperbolic. This is Bers' form of Thurston's trichotomy for mapping classes: finite order, reducible, pseudo-Anosov. This trichotomy also has an interpretation in terms of the action of the mapping class group on the curve complex which is a nonproper Gromov hyperbolic space by a theorem of Masur and Minsky.
For elements of $Out(F_n)$, the outer automorphism group of a rank $n$ free group, there are related trichotomies and other -otomies coming from the work of Bestvina, Feighn, and Handel on relative train track theory. The simplest one is that every element of $Out(F_n)$ is either of finite order, or of polynomial growth, or of exponential growth.