The Infrared Spectrograph (IRS) on the Spitzer Space Telescope observed nearly 800 point sources in the Large Magellanic Cloud (LMC), taking over 1000 spectra. 197 of these targets were observed as part of the SAGE-Spec Spitzer Legacy program; the remainder are from a variety of different calibration, guaranteed time and open time projects. We classify these point sources into types according to their infrared spectral features, continuum and spectral energy distribution shape, bolometric luminosity, cluster membership and variability information, using a decision-tree classification method. We then refine the classification using supplementary information from the astrophysical literature. We find that our IRS sample is comprised substantially of YSO and HII regions, post-main-sequence low-mass stars: (post-)asymptotic giant branch stars and planetary nebulae and massive stars including several rare evolutionary types. Two supernova remnants, a nova and several background galaxies were also observed. We use these classifications to improve our understanding of the stellar populations in the LMC, study the composition and characteristics of dust species in a variety of LMC objects, and to verify the photometric classification methods used by mid-IR surveys. We discover that some widely used catalogues of objects contain considerable contamination and others are missing sources in our sample.