Due to the crucial implication of oil risks for economic growth and policy making, the aim of this paper is to explore the heterogeneous interconnections of supply or demand in oil risks over time horizons and different countries. Specifically, we first examine the correlation of supply or demand in oil return risks and show the relationships in different countries based on wavelet coherence. Furthermore, we explore the time-varying interconnections between supply- or demand-side and oil return risks, as well as oil producers and demand countries. The empirical results show that the correlation between supply and oil return risks is relatively stable, whereas the linkage between demand and oil return risks shows greater volatility due to the impact of specific events. Further study indicates that there are heterogeneous interconnections between supply- or demand-side and oil return risks over sample periods. Specifically, the sign of response could be divided into four phases, i.e., 1997–2002, 2002–2010, 2010–2013 and 2014–2018. In addition, the interconnections of the demand side could be divided into three phases due to the sign of it. What is more, the dynamic interconnections of oil producers’ or countries’ demands behave quite heterogeneously in different countries. Thus policymakers should focus on the coordination level and space capacity in the global crude oil market.