We often use the Cartesian coordinate system, since it is the naive approach at macro level (placing a box just "next to" or "above" the other box). There are, however, many more such systems, incl. these here: http://en.wikipedia.org/wiki/Coordinate_system.
Which of the systems allow modeling experimental data in the most effective way (exploiting the most explicit characteristics of quantum-level interactions)?
There are some subtleties in quantizing Hamiltonians which are in non-cartesian coordinates.
The usual recipe is:
Write down the classical Hamiltonian for your system in terms of canonical coordinates $q_i$ and $p_i$.
Now that you have $H(q_i,p_i)$ you introduce Hermitian operators $Q_i$ and $P_i$ which obey the canonical commutation relation $[Q_i,P_j]=\delta_{ij} i\hbar$.
The Hamiltonian operator is obtained by the substitution $H(Q_i,P_i)$.
It turns out that this recipe sometimes gives the wrong operator when you use canonical coordinates that are not cartesean (for instance see Shankar problem 7.4.10). Part of this stems from the fact the canonical commutation relation is not strict enough to make the choice of operators for $Q_i$ and $P_i$ unique. The moral of the story is that you should quantize in cartesean coordinates before you switch to some other coordinate system.
In general, in physics the question which coordinate system is best suited to approach your problem depends mostly on the symmetry of your problem. If, for example, there is a complete (spatial) rotational symmetry, the obvious choice for where some sort of simplification might be expected is spherical coordinates. The same holds in quantum mechanics: The simple model of the hydrogen atom is a perfect example of a problem that is simpler in spherical coordinates.
