  We study the appearance of notable interval structures -- lattices, modular
lattices, distributive lattices, and boolean lattices -- in both the Bruhat and
weak orders of Coxeter groups. We collect and expand upon known results for
principal order ideals, including pattern characterizations and enumerations
for the symmetric group. This segues naturally into a similar analysis for
arbitrary intervals, although the results are less characterizing for the
Bruhat order at this generality. In counterpoint, however, we obtain a full
characterization for intervals starting at rank one in the symmetric group, for
each of the four structure types, in each of the two posets. Each category can
be enumerated, with intriguing connections to Fibonacci and Catalan numbers. We
conclude with suggestions for further directions and questions, including an
interesting analysis of the intervals formed between a permutation and each
generator in its support.
