Which of the following is the best leaving group in a nucleophilic substitution at a saturated carbon?

A good leaving group is the conjugate base of a strong acid, i.e. a weak, well-stabilised base. Toluenesulfonic acid is a strong acid, while water, methanol and ammonia are extremely weak acids, so $\mathrm{HO}{}^{-}$, $\mathrm{CH}{}_3\mathrm{O}{}^{-}$ and $\mathrm{NH}{}_2{}^{-}$ are strong bases and poor leaving groups. Tosylate is the best.