Fishing impacts on marine food webs are predicted by simulations of a size spectrum community model. In this model, predation is determined by predator and prey size and abundance, and drives predator growth and prey mortality. Fishing amplifies temporal oscillations in the biomass flow. Oscillations appear at lower fishing intensity and have wider amplitude when fishing is selective (removes a narrow size range) and/or when large fish are targeted, than when fishing is more balanced (catching a larger size range) or when small fish are targeted. A novel index of size diversity is developed, and is shown to be sensitive to both fishing intensity and selectivity. To avoid unstable food web dynamics with potential harmful consequences for fisheries, limiting both fishing intensity and selectivity might be an appropriate exploitation strategy.