In positive characteristic, quotient singularities need not be rational. For an example, see Artin's paper "Wildly ramified Z/2-actions in dimension two" (about which I learned from J'anos Koll'ar).

In characteristic zero, quotient singularities are log terminal, and log terminal singularities are rational. The first fact is a simple exercise on discrepancies. For the second one, a very elementary proof is contained in the book of Koll'ar and Mori.

In positive characteristic, quotient singularities need not be rational. For an example, see Artin's paper "Wildly ramified Z/2-actions in dimension two".

In characteristic zero, quotient singularities are log terminal, and log terminal singularities are rational. 

The first fact is a simple exercise on discrepancies (which I think first appeared in a paper of Shokurov), basically a straightforward application of Riemann-Hurwitz ramification formula. 

For the second one, an elementary proof is contained in the book of Koll'ar and Mori.