The driving forces behind cryptoassets’ price dynamics are often perceived as being dominated by speculative factors and inherent bubble-bust episodes. Fundamental components are believed to have a weak, if any, role in the price-formation process. This study examines five cryptoassets with different backgrounds, namely Bitcoin, Ethereum, Litecoin, XRP, and Dogecoin between 2016 and 2022. It utilizes the cusp catastrophe model to connect the fundamental and speculative drivers with possible price bifurcation characteristics of market collapse events. The findings show that the price and return dynamics of all the studied assets, except for Dogecoin, emerge from complex interactions between fundamental and speculative components, including episodes of price bifurcations. Bitcoin shows the strongest fundamentals, with on-chain activity and economic factors driving the fundamental part of the dynamics. Investor attention and off-chain activity drive the speculative component for all studied assets. Among the fundamental drivers, the analyzed cryptoassets present their coin-specific factors, which can be tracked to their protocol specifics and are economically sound.