Overdispersion means more (higher) dispersion than assumed by the model so is a concept that is relative, it depends on the model used. Many (most) models do not assume anything about the dispersion (variance), for instance linear models, which have one (or sometimes more) parameters modeling the variance directly. In such cases (linear models and generalizations anova, linear mixed models, multilevel models ...) there is no possibility of overdispersion, because the variance is modelled directly and independently from the mean.
Overdispersion typically is a possibility when the model do not have a parameter which is directly modeling the variance. The most common cases is binary models like logistic regression where the variance is a function of the mean. The same applies for models based on the multinomial distribution. The other main case is count data models like poisson regression, since the poisson distribution only have one parameter, and the variance equals the mean. But overdispersion is also possible for negative binomial models.
But the main thing to watch out for is models which do not model the variance directly by its own parameter(s).