This study examines the multiscale spillovers and nonlinear causalities between the crude oil futures market and the stock markets of the United States (US), Canada, China, Russia, and Venezuela before and during the COVID-19 pandemic. Using the wavelet coherency method, we find strong co-movement between the oil futures market and these five stock markets, particularly from March 2020 to May 2020 (initial period of the COVID-19 outbreak) at high frequency. Furthermore, we find positive co-movements at low frequency during the overall COVID-19 period. This finding suggests that the bearish trend of stock markets is associated with a downward movement in oil prices. Using the wavelet-based Granger causality approach, we find that the oil and stock indices have less co-movement on a smaller scale but greater movement on a larger scale across all periods. As an exception, the Russian market is significantly influenced by oil prices, even on a small scale, before the COVID-19 period, but not after the beginning of the pandemic. We also find effects in the opposite direction—the Canadian and U.S. markets influence oil prices on a small scale during the COVID-19 period, an effect that is not visible for the U.S. market in the pre-COVID-19 sample. The results also show a significant bidirectional causality from oil to stock markets and vice versa during Russian-Saudi oil price war at high scale. Furthermore, we find that investors should hold more oil futures than stock shares in their portfolios for all periods. This evidence confirms that oil instruments are important for hedging during normal periods and act as safe-haven assets during crisis periods. We observe that the U.S. and Canadian stock markets were more affected by oil price shocks than were other countries.