Multinomial data occur if the major outcome of an experiment is the classification of experimental units into more than two mutually exclusive categories. In experiments with several treatment groups, one may then be interested in multiple comparisons between the treatments w.r.t several definitions of odds between the multinomial proportions. Asymptotic methods are described for constructing simultaneous confidence intervals for this inferential problem. Further, alternative methods based on sampling from Dirichlet posterior distributions with vague Dirichlet priors are described. Monte Carlo simulations are performed to compare these methods w.r.t. their frequentist simultaneous coverage probabilities for a wide range of sample sizes and multinomial proportions: The methods have comparable properties for large samples and no rare events involved. In small sample situations or when rare events are involved in the sense that the expected values in some cells of the contingency table are as low as 5 or 10, the method based on sampling from the Dirichlet posterior yields simultaneous coverage probabilities closest to the nominal confidence level. The methods are provided in an R-package and their application is illustrated for examples from developmental toxicology and differential blood counts.