We present Hubble Space Telescope (HST) optical transmission spectra of the transiting hot-Jupiter WASP-12b, taken with the Space Telescope Imaging Spectrograph instrument. The resulting spectra cover the range 2900-10 300 Å which we combined with archival Wide Field Camera 3 spectra and Spitzer photometry to cover the full optical to infrared wavelength regions. With high spatial resolution, we are able to resolve WASP-12A's stellar companion in both our images and spectra, revealing that the companion is in fact a close binary M0V pair, with the three stars forming a triple-star configuration. We derive refined physical parameters of the WASP-12 system, including the orbital ephemeris, finding the exoplanet's density is ∼20 per cent lower than previously estimated. From the transmission spectra, we are able to decisively rule out prominent absorption by TiO in the exoplanet's atmosphere, as there are no signs of the molecule's characteristic broad features nor individual bandheads. Strong pressure-broadened Na and K absorption signatures are also excluded, as are significant metal-hydride features. We compare our combined broad-band spectrum to a wide variety of existing aerosol-free atmospheric models, though none are satisfactory fits. However, we do find that the full transmission spectrum can be described by models which include significant opacity from aerosols: including Rayleigh scattering, Mie scattering, tholin haze and settling dust profiles. The transmission spectrum follows an effective extinction cross-section with a power law of index α, with the slope of the transmission spectrum constraining the quantity αT = -3528 ± 660 K, where T is the atmospheric temperature. Rayleigh scattering (α = -4) is among the best-fitting models, though requires low terminator temperatures near 900 K. Sub-micron size aerosol particles can provide equally good fits to the entire transmission spectrum for a wide range of temperatures, and we explore corundum as a plausible dust aerosol. The presence of atmospheric aerosols also helps to explain the modestly bright albedo implied by Spitzer observations, as well as the near blackbody nature of the emission spectrum. Ti-bearing condensates on the cooler night-side is the most natural explanation for the overall lack of TiO signatures in WASP-12b, indicating the day/night cold trap is an important effect for very hot Jupiters. These findings indicate that aerosols can play a significant atmospheric role for the entire wide range of hot-Jupiter atmospheres, potentially affecting their overall spectrum and energy balance.