ABSTRACT When ordinal responses to questionnaires structured on the basis of Likert scales show differing variability or heterogeneity in subgroups of the population, appropriate regression approaches that are able to take this issue into account are the location‐scale and location‐shift model. If data come in clusters, which causes within‐cluster variance, an additional cluster‐level random effect specification is due. Cumulative models for ordinal responses are considered assessing the responses in terms of mean level (or location), variability (or scale), heterogeneity (or dispersion) and in terms of random effects related to clusters. Furthermore, in order to reduce the complexity of the models, a variable selection procedure through adaptive fused LASSO‐type regularization is proposed. A case study with data from the Survey of Health, Ageing and Retirement in Europe is used to demonstrate the applicability of the models and the properties of the selection procedures. It is shown that variable selection by regularization produces stable parameter estimates and results that are easy to interpret in all model components. The performance of the proposed regularization approach is further assessed by means of a simulation study.