We present CCD phase-binned light curves at 490nm for 97 Cepheid variable stars in the bar of the LMC. The photometry was obtained as part of the French EROS project and has excellent phase coverage, permitting accurate decomposition into Fourier components. We identify as `sinusoidal' or s-Cepheids those stars with periods less than 5.5d and small second-harmonic components. These stars comprise ~30% of our sample and most form a sequence ~1mag brighter than the LMC classical Cepheids in the period-luminosity diagram. They are also generally bluer and have lower-amplitude light curves. We infer that the s-Cepheids are first-overtone pulsators because, when their periods are converted to expected fundamental-mode values, they obey a common period-luminosity-colour relation with classical Cepheids. This also confirms the reality of the colour term in the Cepheid period-luminosity-colour relation. Further, the blue edge of the classical Cepheid instability strip agrees well with the theoretical calculations for the fundamental mode made by Chiosi et al. 1993 (=1993ApJS...86..541C) for the Hertzsprung-Russell and period-luminosity diagrams, but we find that our observed s-Cepheids are >0.2mag brighter and bluer than the Chiosi et al. predictions for the first-overtone. We identify a number of features in plots of our stars' Fourier-component amplitude ratios and phase differences. These features have been identified with resonances between different pulsation modes. In the LMC we find these features seem to occur at periods very similar to Galactic ones for classical Cepheids, but at different periods for s-Cepheids. We discover a double-mode Cepheid in the LMC, for which P(first overtone)/P(fundamental)=0.710+/-0.001, very similar to observed ratios for Galactic double-mode Cepheids.