It is well-known that log terminal singularities are rational, but log canonical singularities are not. On the other hand, rational singularities are not necessarily $\mathbb Q$-Gorenstein, so there are rational singularities that are not even log canonical (and hence not log terminal). On the third hand (if someone has one), rational Gorenstein singularities are canonical and hence log terminal and hence log canonical.

So that leads to the question: Is there a rational singularity that is $\mathbb Q$-Gorenstein, but not log canonical?