I'm having a hard time to find a good (and modern) classification scheme for phase transitions and related universality classes. Can someone recommend a paper/book/site? Detailed mathematical aspects are very much welcome.
I am afraid that your question is too ambitious. It's almost like asking "where I can find a good resource of a theory of everything, including all emergent and complex phenomena in the world?". The question has many aspects.
Different physical systems have different numbers of different phases. In principle, there are infinitely many types of physical systems, so there are also infinitely many possible phase diagrams.
It is good to know a sufficient number of phase transitions - as well as their basic theories such as Landau's theory of second-order phase transitions - but it's hard to claim that one already knows "all of them" at a given moment - which is needed for claiming that they have been fully classified.
Incidentally, it's not always the case that the number of inequivalent phase transitions is increasing. Sometimes, it may be the other way around. For example, the AdS/CFT correspondence has shown that various phase transitions in gravitational theories - such as the Hawking-Page transition - are completely equivalent to some transitions in non-gravitational theories - such as the confinement/deconfinement transition.
This is the insight of the last decade or so. Similar insights are being added even today. The science of phase transitions hasn't been closed. Phase transitions are pretty generic properties of most physical systems - so whenever one learns new things about some physical systems, it's likely that one also learns something about some of their phase transitions.
Depending on what you really want, you may choose one of the books about phase transitions, e.g. from this list:
Best wishes Lubos
The basic classification of phase transitions is into first and second order transitions. The former is characterized by having a nonzero latent heat, e.g. the liquid-vapor transition of $H_2 O$. Often one can adjust a thermodynamic parameter to construct a line of first order transitions and this line terminates at a critical point where there is a second order transition. Second order transitions are characterized by an infinite correlation length (in field theory terms the mass parameter goes to zero). Critical points are often studied using the tools of Conformal Field Theory. The critical exponents which govern behavior near the critical point depend on the universality class which often depends on the symmetry of the system. The Wikipedia page http://en.wikipedia.org/wiki/Phase_transition will give you a quick overview. I recall that the book "Modern Theory of Critical Phenomenon" by S. Ma has more detailed and pedagogical discussion of these matters.
If you already know something about phase transitions and universality classes and want to know what kind of phenomena are similar to each other, I would recommend the following article:
Géza Ódor, Universality classes in nonequilibrium lattice systems, REVIEWS OF MODERN PHYSICS, VOLUME 76, JULY 2004
This contains lots of different systems that can be (are) modeled with lattices (e.g. Ising, percolation and interface growth). The contents list is actually pretty good starting point for simple listed classification.