We investigate the effects of a set of regulatory policies directed towards high-frequency trading (HFT) through an agent-based model of a limit order book able to generate flash crashes as the result of the interactions between low- and high-frequency traders. In particular, we study the impact of the imposition of minimum resting times, of circuit breakers, of cancellation fees and of transaction taxes on asset price volatility and on the occurrence and the duration of flash crashes. Monte-Carlo simulations reveal that HFT-targeted policies imply a trade-off between market stability and resilience. Indeed, we find that policies able to tackle volatility and flash crashes also hinder the market from quickly recovering after a crash. This result is mainly due to the dual role of HFT, as both a cause of flash crashes and a key player in the post-crash recovery.