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We study the appearance of notable interval structures -- lattices, modularlattices, distributive lattices, and boolean lattices -- in both the Bruhat andweak orders of Coxeter groups. We collect and expand upon known results forprincipal order ideals, including pattern characterizations and enumerationsfor the symmetric group. This segues naturally into a similar analysis forarbitrary intervals, although the results are less characterizing for theBruhat order at this generality. In counterpoint, however, we obtain a fullcharacterization for intervals starting at rank one in the symmetric group, foreach of the four structure types, in each of the two posets. Each category canbe enumerated, with intriguing connections to Fibonacci and Catalan numbers. Weconclude with suggestions for further directions and questions, including aninteresting analysis of the intervals formed between a permutation and eachgenerator in its support.

Interval structures in the Bruhat and weak orders