For the QHE you need a 2 dimensional electron gas and high magnetic fields and low temperature. The band structure that gives rise to the formation of a 2d electron gas can  be tailored, e.g., in semiconductor heterostructures where the equilibrium condition requires matching of the Fermi energy aka chemical potential (Gibbs free energy per particle) creates a triangular shaped band. So the choice is made by nature not so much by the experimentalist. Having that said, the experimentalist needs to be aware that other conditions, such as high magnetic fields are needed to see the quantization effects. 

Ok, lets answer your question by parts. As you said, the classical Hall effect is a very efficient way to calculate the sign of the charge carriers and the charge carries density in metals.  So, if that is what you want to do you just need a regular magnetic field, a metal bar of reasonable quality and a multimeter and your are done. On the other hand, one of the first practical use of the Quantum Hall effect was to measure the resistivity quantum, which later on was used to defined a new standard for electrical resistance (see art. http://iopscience.iop.org/0034-4885/64/12/201/ )  the quantum Hall effect can also be used to determine the Fine-structure constant with great acuracy. So, as you can see, for practical application, the quantum hall effect can be used for metrology.