The paper focuses on severe housing deprivation and its components—overcrowding and housing conditions problems. It is based on the European Union Statistics on Income and Living Conditions (EU-SILC) 2019 survey data. The paper analyzes this data on households in Poland—a country with significant problems in terms of housing deprivation. Three dimensions of housing deprivation are examined: experiencing only overcrowding, experiencing only housing conditions problems, and being severely housing deprived. The study aims to investigate three-dimensional housing deprivation depending on the urbanization level of living places and other socioeconomic characteristics of Polish households. The multinomial logit model was used to assess relative risk ratios for explanatory variables to achieve these purposes. It was found that some socioeconomic characteristics significantly influenced one dimension but not another. Moreover, the relative risk ratios for given characteristics sometimes are greater than one for a particular dimension and less than one for another. Specifically, a significant difference in housing conditions problems between households living in towns and rural areas was found. However, there are no significant differences in overcrowding and severe housing deprivation, given that all other explanatory variables are fixed. Furthermore, considering the relative risk as a ratio of the probability of experiencing the given dimension of housing deprivation and the probability of not being housing deprived at all, households in cities compared with rural households were more likely to be overcrowded and to experience severe housing deprivation but less likely to have housing conditions problems.