Surfactants are commonly used as disinfection agents in personal care products against bacteria and viruses, including SARS-CoV-2. However, there is a lack of understanding of the molecular mechanisms of the inactivation of viruses by surfactants. Here, we employ coarse grain (CG) and all-atom (AA) molecular dynamics simulations to investigate the interaction between general families of surfactants and the SARS-CoV-2 virus. To this end, we considered a CG model of a full virion. Overall, we found that surfactants have only a small impact over the virus envelope, being inserted into the envelope without dissolving it or generating pores, at the conditions considered here. However, we found that surfactants may induce a deep impact on the spike protein of the virus (responsible for its infectivity), easily covering it and inducing its collapse over the envelope surface of the virus. AA simulations confirmed that both negatively and positively charged surfactants are able to extensively adsorb over the spike protein and get inserted into the virus envelope. Our results suggest that the best strategy for the design of surfactants as virucidal agents will be to focus on those strongly interacting with the spike protein.