How to determine the mass of a quark? As far as I know quarks are never found in isolation, so how can we determine their rest mass?
 A: For the light quarks, one can use chiral perturbation theory to relate the mass of the light hadrons to the mass of the light quarks. These two links give details and caveats of the procedures, as well as the most precise determinations:
http://pdg.lbl.gov/2011/reviews/rpp2011-rev-quark-masses.pdf
http://pdg.lbl.gov/2011/listings/rpp2011-list-light-quarks.pdf
A: A couple of relevant issues bear noting:


*

*It is hard to determine the masses of the confined quarks because the equations of QCD that are used to determine them have infinite series solutions (like almost all Standard Model equations based upon path integral propagators) that converge much, much more slowly than the comparable equations for the electromagnetic force of QED and the weak force. We can measure the mass of the proton to nine significant digits, but can only calculate it to three using QCD and even three digit precision requires literally months and even years of computer assisted calculations.


Sometimes we can calculate relationships between two calculations (e.g. between the up and down quark mass) because that particular relationship depends almost entirely on QED relationships that are easy to calculate.  But, in general, it is a difficult problem.


*There are definitional issues that go into deciding what counts as the mass of a light quark (i.e. up, down or strange), because those masses are below characteristic QCD energy scale which is about 330 MeV.


Quark masses "run" with the energy scale of the interaction they are measured in, in much the same way that each of the Standard Model coupling constants do, as a non-trivial physical consequence of the mathematical trick called renormalization which is necessary to get sensible results out of Standard Model equations. Thus, the mass of a bottom quark in a 5 GeV momentum transfer interaction is different than the mass of a bottom quark in a 50 GeV momentum transfer interaction.
In the case of heavy quarks, this makes it possible to define quark masses in a "natural" way that is insensitive to the renormalization scheme involved by looking at the "pole mass" of the quark (i.e. its mass at an energy scale equal to its own mass).
In the case of the light quarks, however, "pole masses" are ill defined because light quarks never appear outside of hadrons that are much more massive than the quarks themselves. So, instead of "pole masses", the light quark masses are typically evaluated under a particular renormalization scheme such as the MS one at an energy scale of 1 GeV or 2 GeV corresponding to one or two nucleons, the context in which they are most commonly observed.
Naive extrapolations of renormalization formulas from perturbative QCD (which is outside of the usual range of applicability of perturbative QCD) produce the absurd result that the light quark masses sometimes exceed the masses of some mesons like the pion in which light quarks are present.
It isn't entirely clear whether the "pole mass" of a light quark is merely impractical to determine or requires non-perturbative methods that haven't been worked out properly yet, or if the absurd results that flow from naive attempts to determine a "pole mass" for a light quark are actually a strong hint that pole mass is a fundamentally flawed concept in this context that does not apply to light quarks.
There is no "practical" reasons that we need the "pole mass" of a liqht quark to do physics. All relevant quark masses that we need to know can be determined in the MS scheme (or some of the alternatives) which can allow us to do the calculations that need to be done. 
But, in many efforts to look for deeper meaning and more fundamental relationships from a theory that the Standard Model might be only a low energy effective field theory of, you need to be able to define the physical constants of the model in a consistent and "natural" way in order to be able to meaningfully evaluate whether or not there are mathematical relationships between them or not.
For example, if there were some relationship between the top quark mass and the up quark mass that only holds if you are consistently using "pole masses" across the board, you might be led astray and find yourself unable to see the relationship if you used the MS mass of the up quark evaluated at 1 GeV instead.
