The evolution of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has highlighted the need for versatile diagnostic assays that can discriminate among emerging variants of the virus. Here we report the development and performance benchmarking of an inexpensive (approximately US$0.30 per test) assay for the rapid (sample-to-answer time within 30 min) colorimetric detection of SARS-CoV-2 variants. The assay, which we integrated into foldable paper strips, leverages nucleic acid strand-displacement reactions, the thermodynamic energy penalty associated with single-base-pair mismatches and the metal-ion-controlled enzymatic cleavage of urea to amplify the recognition of viral RNAs for the colorimetric readout of changes in pH via a smartphone. For 50 throat swab samples, the assay simultaneously detected the presence of SARS-CoV-2 and mutations specific to the SARS-CoV-2 variants Alpha, Beta and Gamma, with 100% concordance with real-time quantitative polymerase chain reaction and RNA sequencing. Customizable and inexpensive paper-based assays for the detection of viruses and their variants may facilitate viral surveillance.