Aims. We present a comprehensive study on the supernova remnant (SNR) population of the Small Magellanic Cloud (SMC). We measured multiwavelength properties of the SMC SNRs and compare them to those of the Large Magellanic Cloud (LMC) population. Methods. This study combines the large dataset of XMM-Newton observations of the SMC, archival and recent radio continuum observations, an optical line emission survey, and new optical spectroscopic observations. We were therefore able to build a complete and clean sample of 19 confirmed and four candidate SNRs. The homogeneous X-ray spectral analysis allowed us to search for SN ejecta and Fe K line emission, and to measure interstellar medium abundances. We estimated the ratio of core-collapse to type Ia supernova rates of the SMC based on the X-ray properties and the local stellar environment of each SNR. Results. After the removal of unconfirmed or misclassified objects, and the addition of two newly confirmed SNRs based on multi-wavelength features, we present a final list of 21 confirmed SNRs and two candidates. While no Fe K line is detected even for the brightest and youngest SNR, we find X-ray evidence of SN ejecta in 11 SNRs. We estimate a fraction of 0.62–0.92 core-collapse supernova for every supernova (90% confidence interval), higher than in the LMC. The difference can be ascribed to the absence of the enhanced star-formation episode in the SMC, which occurred in the LMC 0.5–1.5 Gyr ago. The hot-gas abundances of O, Ne, Mg, and Fe are 0.1–0.2 times solar. Their ratios, with respect to SMC stellar abundances, reflect the effects of dust depletion and partial dust destruction in SNR shocks. We find evidence that the ambient medium probed by SMC SNRs is less disturbed and less dense on average than in the LMC, consistent with the different morphologies of the two galaxies.