Defining precisely what are all the quantum numbers is a difficult question because it depends highly on the model under consideration, even for the standard model.
In particular any U(1) symmetry leads to a quantum number, and similarly some U(1) subgroup of non-abelian groups that commute with all other interactions can also be associated to quantum numbers (such as the weak isospin). So the more your model has, the more numbers you will have.
Now even if you ask the question for the "real" world described by the Standard Model it is not clear how to count these numbers: this depends highly on the precise domain you are studying. For example if you are looking just as the quarks u, d, and s, then the "strangeness" is a good quantum number, but it is not anymore when you add weak interaction (like other flavour numbers). Similarly the conservation of the individual lepton numbers hold well and is useful for many purposes, but it is not exact due to neutrino oscillations.
Finally our current classification of quantum numbers could change if new interactions were discovered: for example some unification theories predict that the proton can decay and this would yield a violation of the baryon number.
So here is the list of the quantum numbers, in the context of the Standard model, according to the current status:
- true charges: electric charge (related to the weak hypercharge and isospin), baryon and lepton numbers (the latter including subcases;
- flavour charges (violated by weak interactions): one for each quark (strange, charm, bottomn, top), isospin;
- leptonic charges (violated by neutrino oscillations): electronic, muonic and tauonic lepton numbers.