Previous MO questions:

 * rational/trigonometric/elliptic trichotomy
https://mathoverflow.net/questions/58040/groups-quantum-groups-and-fill-in-the-blank

 * trichotomy of interrelated model structures: h-model, q-model, m-model
https://mathoverflow.net/questions/86942/is-the-category-of-metric-spaces-and-continuous-maps-quillen-equivalent-to-top

 * "such "log-exp functions" are either eventually positive, eventually zero, or eventually negative. … It guarantees that the germs at infinity of such functions do indeed form a field K."
https://mathoverflow.net/questions/45284/examples-of-sequences-whose-asymptotics-cant-be-described-by-elementary-function

 * a function of a complex variable with an algebraic addition theorem must be: 1) A rational function, 2) A rational function of e^{px}, or 3) A rational function of the Weierstrass elliptic function and its derivative.
https://mathoverflow.net/questions/96452/trig-functions-based-on-convex-curves

 * "Every finitely generated infinite profinite group has a just infinite quotient. There is a trichotomy due to Wilson (and refined by Grigorchuk) describing what they can look like."
https://mathoverflow.net/questions/49591/what-is-the-virtue-of-profinite-groups-as-mathematical-objects/68895

 * "there is a trichotomy of curves given by g=0, g=1, and g≥2. If you look at topological, geometric, arithmetic properties of these curves, their properties align very strongly with these classes."
https://mathoverflow.net/questions/56011/why-should-i-believe-the-mordell-conjecture

 * Kodaira dimension. κ(Y)<0, κ(Y)=0, κ(Y)=dimY.
https://mathoverflow.net/questions/81913/how-frequent-are-smooth-projective-varieties-with-anti-ample-canonical-bundle

 * "Rank and period of primes in the Fibonacci sequence; a trichotomy," Fib. Quart., 45 (No. 1, 2007), 56-63).
https://mathoverflow.net/questions/84797/can-the-difference-of-two-distinct-fibonacci-numbers-be-a-square-infinitely-often 

 * According to Etingof, Igor Frenkel once suggested that there are three "levels" to Lie theory, which could be called no loops, one loop, and two loops.
https://mathoverflow.net/q/186339/2051

M.SE:

* "The set-theoretic setup of Categories for the working mathematician is somewhat subtle. … There is therefore a trichotomy of small sets, large sets, and proper classes. This is not the usual practice: we normally think of all sets as being small."
[Answer by Zhen Lin](https://math.stackexchange.com/a/201188) to [confusion over the use of universes in category theory](https://math.stackexchange.com/questions/201062/confusion-over-the-use-of-universes-in-category-theory)

* "There are three distinct aspects of schemes that each have their own purpose …."
[Answer by Michael Joyce](https://math.stackexchange.com/a/99615) to https://math.stackexchange.com/questions/99605/why-study-schemes/99615

TCS.SE:

* "… one of the most amazing facts about logic is that consistency strength boils down to the question "what is the fastest-growing function you can prove total in this logic?" As a result, the consistency of many classes of logics can be linearly ordered! If you have an ordinal notation capable of describing the fastest growing functions your two logics can show total, then you know by trichotomy that either one can prove the consistency of the other, or they are equiconsistent."
[Answer by Neel Krishnaswami](https://cstheory.stackexchange.com/a/4821) to [Axioms necessary for theoretical computer science](https://cstheory.stackexchange.com/questions/4816/axioms-necessary-for-theoretical-computer-science/4821)


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A frequently cited paper: Caicedo and Ketchersid, "[A trichotomy theorem in natural models of AD+](http://dx.doi.org/10.1090/conm/533/10510)", in "Set Theory and Its Applications", Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 227–258.