Vector spaces with natural bases Sergeib's question asks about vector spaces without a natural basis.
Actually, I would claim (apparently in accord with many comments and answers to Sergeib's question ) that this is the default situation and that it is a rare event if a naturally occurring vector space does have a natural basis or even if it has any basis that can be described explicitly: the notorious $\mathbb Q \; -$ vector space $\mathbb R$ being the foremost example of this impossibility of exhibiting an explicit basis . Of course  tautological examples like polynomial rings don't contradict this thesis, since they are defined as free vector spaces on some set!
In fact, the only example I can see without thinking twice of a natural vector space  with a big natural non-tautological basis is the rational function field $k(X)$ seen as a $k$ - vector space .
Namely, consider a field $k$ and the set of monic polynomials  $f_i(X) \; (i \in I)$ irreducible over $k$. Then  $k(X)$ has the following $k$ - basis:
the mononomials $X^k (k\in \mathbb N) $ and the rational fractions $\frac{X^m}{f_i(X)^s}$
(with $i\in I,\; s>0$ and $ m< deg f_i $ )
In particular this natural basis is non denumerable if $k$ is non denumerable..
Question  Which other vector spaces do you know for which some (preferably big) explicit basis can be given, but which are not clearly constructed as free vector spaces over a set? 
 A: I'll echo Ben Webster's comment from the other thread, but in the other direction: find some natural commuting operators whose simultaneous eigenspaces are all one-dimensional.  This is how, for example, one gets the root space decomposition in Lie theory and the decomposition of spaces of modular forms into eigenforms.  (Admittedly, the former is only unique up to a choice of Cartan subalgebra.)
I guess that to get an actual basis instead of a basis-up-to-scalar-multiplication one needs a little more.  For cusp forms one usually takes the normalization where the coefficient of $q$ in the Fourier expansion is $1$, whereas for the Lie algebra I guess we take a normalization where the structure constants are as nice as possible.
A: 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. 
A: $H^*(\mathbb{CP}^{\infty},k) = k[x]$ has basis $\{x^i : i \geq 0\}$. Of course, every other nontrivial computation of (co)homology groups fits in here.
