Variable selection and Bayesian effect fusion for categorical predictors in linear regression models. Effect fusion aims at the question which categories have a similar effect on the response and therefore can be fused to obtain a sparser representation of the model. Effect fusion and variable selection can be obtained either with a prior that has an interpretation as spike and slab prior on the level effect differences or with a sparse finite mixture prior on the level effects. The regression coefficients are estimated with a flat uninformative prior after model selection or model averaged. For posterior inference, an MCMC sampling scheme is used that involves only Gibbs sampling steps.