Background Infection with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) induces an antibody response targeting multiple antigens that changes over time. This study aims to take advantage of this complexity to develop more accurate serological diagnostics. Methods A multiplex serological assay was developed to measure IgG and IgM antibody responses to seven SARS-CoV-2 spike or nucleoprotein antigens, two antigens for the nucleoproteins of the 229E and NL63 seasonal coronaviruses, and three non-coronavirus antigens. Antibodies were measured in serum samples collected up to 39 days after symptom onset from 215 adults in four French hospitals (53 patients and 162 health-care workers) with quantitative RT-PCR-confirmed SARS-CoV-2 infection, and negative control serum samples collected from healthy adult blood donors before the start of the SARS-CoV-2 epidemic (335 samples from France, Thailand, and Peru). Machine learning classifiers were trained with the multiplex data to classify individuals with previous SARS-CoV-2 infection, with the best classification performance displayed by a random forests algorithm. A Bayesian mathematical model of antibody kinetics informed by prior information from other coronaviruses was used to estimate time-varying antibody responses and assess the sensitivity and classification performance of serological diagnostics during the first year following symptom onset. A statistical estimator is presented that can provide estimates of seroprevalence in very low-transmission settings. Findings IgG antibody responses to trimeric spike protein (Stri) identified individuals with previous SARS-CoV-2 infection with 91·6% (95% CI 87·5–94·5) sensitivity and 99·1% (97·4–99·7) specificity. Using a serological signature of IgG and IgM to multiple antigens, it was possible to identify infected individuals with 98·8% (96·5–99·6) sensitivity and 99·3% (97·6–99·8) specificity. Informed by existing data from other coronaviruses, we estimate that 1 year after infection, a monoplex assay with optimal anti-Stri IgG cutoff has 88·7% (95% credible interval 63·4–97·4) sensitivity and that a four-antigen multiplex assay can increase sensitivity to 96·4% (80·9–100·0). When applied to population-level serological surveys, statistical analysis of multiplex data allows estimation of seroprevalence levels less than 2%, below the false-positivity rate of many other assays. Interpretation Serological signatures based on antibody responses to multiple antigens can provide accurate and robust serological classification of individuals with previous SARS-CoV-2 infection. This provides potential solutions to two pressing challenges for SARS-CoV-2 serological surveillance: classifying individuals who were infected more than 6 months ago and measuring seroprevalence in serological surveys in very low-transmission settings. Funding European Research Council. Fondation pour la Recherche Médicale. Institut Pasteur Task Force COVID-19.