The COVID-19 pandemic poses a serious threat to investors in the crude oil market. Furthermore, investors have an increasing need to find a safe haven in their investment portfolios when facing unprecedented risks in crude oil markets during the COVID-19 pandemic. According to a review of the literature, there are contradictory findings on which investment is the safer haven for the oil market. Therefore, this paper aims to evaluate whether bitcoin is a safer haven for the crude oil market than the commonly used gold during the COVID-19 pandemic. Three spillover measurements based on the time, and frequency domains, and a network framework are employed to quantify the return spillover effects among bitcoin, gold and three major crude oil futures markets. We divide the sample into two periods, pre-COVID-19 and post-COVID-19. The results show that bitcoin has a weak safe-haven effect on the crude oil market only over a short period, while gold maintains a good safe-haven ability for crude oil futures across various time horizons (frequencies), both before and after the outbreak of the COVID-19 pandemic. The findings of this study have important implications for policy-makers, crude oil producers and global investors. In particularly, investors cannot ignore the importance of bitcoin and gold in selecting more profitable portfolio policies when searching for safe-haven assets.