Unboundedness of the likelihood typically happens when a class involves only a single individual, giving zero variance in the denominator. But this tends to occur when a model has class-specific variances - the Mplus default is class-invariant variances. In any case, such a solution is clearly an anomolous one on the edge of the parameter space, and not of interest. With many random starts, replicated solutions in the interior of the parameter space are likely.
Yes, local solutions are prevalent also with 2 classes. Mplus gives a warning implying that the default of 10 random starts may not be sufficient when there are more than 2 classes, but already with 2 classes more random starts may well be needed.