The trading market of carbon emission permits is not only an effective tool for tackling climate change, but it also reinforces the connection between carbon and energy markets by risk transmission. This paper investigates the volatility spillover effects and dynamic correlation between fossil energy, clean energy and the European Union carbon markets by applying the asymmetric BEKK- and DCC-GARCH models in a time-frequency framework and designs the portfolio strategies. Empirical results display that fossil energy market has a significantly positive impact on the clean energy market at most scales, but the feedback transmission is relatively weak. Two indirect volatility spillover effects exist between the three markets, which are ‘fossil energy→carbon→clean energy’ and ‘clean energy→carbon→fossil energy’, suggesting that the carbon market plays a vital role in bridging the clean and fossil energy markets. Significant time-varying asymmetric effects are identified between the three markets, and the carbon market is confirmed to offer the least expensive hedge to an investment in oil. Our findings can provide meaningful implications for policymakers to design market mechanism and investors to adjust hedging strategies and diversify their portfolios.