High-temperature anomalies (HTAs) of the earth's surface, such as fires, volcanic activities, and industrial heat sources, have a profound impact on Earth's system. Sentinel-2 Multispectral Instrument (MSI) provides spatially-specific information for precisely measuring the location and extent of HTAs at a fine scale. However, detecting HTAs from MSI images remains challenging because the emitted radiance of an HTA in the short-wave infrared (SWIR) bands can be easily mixed with the reflected solar radiance background in the daytime; and an increasing number of atypical cases in MSI images need to be treated with the enhanced spatial resolution. A generic HTA detection approach that handles both anthropogenic and natural HTAs will broaden the scope of MSI applications. In this study, (i) we highlight two spectral characteristics of HTAs in the far-SWIR, near-SWIR, and NIR bands (i.e., (ρfar-SWIR - ρnear-SWIR)/ρNIR ≥ 0.45 and (ρfar-SWIR -ρnear-SWIR) ≥ ρnear-SWIR - ρNIR) that can effectively enhance HTAs from background geo-features, based on the reflectance spectra in airborne imaging spectrometer data. (ii) We propose a tri-spectral thermal anomaly index (TAI) that jointly uses the two high-temperature-sensitive SWIR bands and the high-temperature-insensitive NIR band to enhance HTAs, based on the above characteristics and a comprehensive sampling of different types of HTAs from 1,974 MSI images. (iii) We develop a TAI-based approach for MSI images to detect HTAs in general. The proposed approach was applied to detect different types of HTAs, including different biomass burnings, active volcanoes, and industrial HTAs, over a wide range of land-cover scenarios. Validations and comparisons demonstrate the proposed approach is reliable and performs better than the existing state-of-the-art HTA detection approaches. Evaluations on two types of small industrial HTAs, including operating kilns and enclosed landfill gas flares, show that the HTA detection probability of the TAI-based approach from time-series MSI images is ~ 84.91% and 88.23%, respectively. Further investigations show that the TAI-based approach also has good transferability in detecting HTAs from multispectral images acquired by Landsat-family satellites.