We present a novel approach to measure the attenuation curves of 485 individual star-forming galaxies with M-* > 10(10) M-circle dot based on deep optical spectra from the VLT/VIMOS LEGA-C survey and multiband photometry in the COSMOS field. Most importantly, we find that the attenuation curves in the rest-frame 3000-4500 angstrom range are typically almost twice as steep as the Milky Way, LMC, SMC, and Calzetti attenuation curves, which is in agreement with recent studies of the integrated light of present-day galaxies. The attenuation at 4500 angstrom and the slope strongly correlate with the galaxy inclination: face-on galaxies show less attenuation and steeper curves compared to edge-on galaxies, suggesting that geometric effects dominate observed variations in attenuation. Our new method produces 2175 angstrom UV bump detections for 260 individual galaxies. Even though obvious correlations between UV bump strength and global galaxy properties are absent, strong UV bumps are most often seen in face-on, lower-mass galaxies (10 < log(10)(M-*/M-circle dot) < 10.5) with low overall attenuation. Finally, we produce a typical attenuation curve for star-forming galaxies at z similar to 0.8; this prescription represents the effect of dust on the integrated spectral energy distributions of high-redshift galaxies more accurately than commonly used attenuation laws.