This study combines the variational mode decomposition (VMD) method and static and time-varying symmetric and asymmetric copula functions to examine the dependence structure between crude oil prices and major regional developed stock markets (S&P500, stoxx600, DJPI and TSX indexes) during bear, normal and bull markets under different investment horizons. Furthermore, it analyzes the upside and downside short- and long-run risk spillovers between oil and stock markets by quantifying three market risk measures, namely the value at risk (VaR), conditional VaR (CoVaR) and the delta CoVaR (∆CoVaR). The results show that there is a tail dependence between oil and all stock markets for the raw return series. By considering time horizons, we show that there is an average dependence between the considered markets for the short-run horizons. However, the tail dependence is also found for the long-run horizons between the oil and stock markets, with the exception of the S&P500 index which exhibits average dependence with the oil market. Moreover, we find strong evidence of up and down risk asymmetric spillovers from oil to stock markets and vice versa in the short-and long run horizons. Finally, the market risk spillovers are asymmetric over the time and investment horizons.