This study uses explainable artificial intelligence to analyze major predictors of poor self-rated health, dementia and their comorbidity, which include various aspects of social support. Data came from the Korean Longitudinal Study of Ageing (2016-2018), with 5527 participants aged 56 or more. The dependent variables were: (1) poor self-rated health (SRH) with 5 categories (very good, good, middle, poor, very poor); (2) dementia (no, yes); and (3) their comorbidity with 4 categories (SRH very good, good or middle, dementia no; SRH very good, good or middle, dementia yes; SRH poor or very poor, dementia no; SRH poor or very poor, dementia yes). The 49 demographic, socioeconomic and health-related predictors were included. The accuracy and area under the receiver operating characteristic curve (AUC) were compared for logistic regression and the random forest. Logistic regression and the random forest registered similar AUCs with the range of 0.80-0.87. Based on random forest variable importance, 18 out of top-20 predictors were identical for SRH, dementia and their comorbidity: previous SRH, life satisfaction - health, age, body mass index, income, chronic diseases, life satisfaction - economic, life satisfaction - overall, grandchildren alive, friendship meeting, brothers/sisters alive, class identity, children alive, social activity - friendship, education, children in weekly contact, religion and drinker. However, some variations were found among the three dependent variables, i.e., previous SRH for comorbidity (1st) and SRH (1st) compared to dementia (11th), grandchildren alive for SRH (6th) and dementia (4th) compared to comorbidity (9th), chronic diseases for comorbidity (6th) compared to SRH (10th) and dementia (10th). In terms of Shapley Additive Explanation (SHAP) values, the probability of dementia is expected to decrease by 5.57% in case a predictor grandchildren alive is included to the model. This study highlights the importance of social support for the prediction of poor self-rated health, dementia and their comorbidity. Specifically, the promotion of family support and friendship activity for elders would be vital for the prevention and management of their self-rated health and dementia.