I somewhat disagree with the premise of the question: I want to point out that after imposing more structure, this also becomes a default situation. The general setting I have in mind is commutative algebra and there are various techniques to construct a "preferred" SAGBI basis in the coordinate ring or homogeneous coordinate ring of a variety. Particularly useful in representation theory are standard monomial bases in $K[G/U],$ where $G$ is a semisimple algebraic group and $U$ is the unipotent radical of a parabolic subgroup.

Gröbner bases of various kinds are of similar ilk. Finally, one should mention orthogonal polynomials: starting with an "obvious" basis, perhaps a naive monomial basis in some polynomial algebra, produce a different standard basis.