Which quasiparticles follow which statistics Let me say beforehand that I know this is an ill-defined question, but I believe it is useful anyway. 
For these common quasiparticles: 


*

*Phonons

*Holes

*Plasmons

*Excitons

*Plasmon-polaritons


What statistics do they follow, and under which assumptions? What would be a general way to determine the statistics for a given quasi-particle?
 A: It depends if we look at particle as classical ones or quantum ones. In the first case, particles are usually following a Boltzmann statistics. However, things become more interesting when entering the quantum world. Here, the spin of the particles become crucial. We have that particles with integer spin follow a Bose-Einstein statistics. Whereas particles with half-integer spin follow a Fermi-Dirac statistics. The same is also true if we look at quasi-particle, also in this case we can clearly assign a label for the type of spin. The role of statistic in the quantum world is associated with the exchange of two particles. For Bosons, nothing is changed to the wave-function after the exchange whereas for Fermions the wave-function is multiplied by a minus sign. 
An important generalization of the concept of quantum statistic was obtained with the discovery of the fractional quantum Hall effect. Under the action of a strong magnetic field, the quasi particles composed by electrons and flux quanta of the magnetic field, also called composite fermions, can acquire a statistic that is not a Bose-like or a Fermi-like. In this case we speak about anyons. Under the exchange of two anyons the wave-function can be multiplied by a complex number. This is very interesting because it paves the way to braiding these particles in order to perform operations that leads to quantum computing. 
A lot of effort is done nowadays because Majorana quasi-particles have an anyonic statistic. Therefore their manipulation could lead to the so called topological quantum computation.
