Reliable forecasts for the dispersion of oceanic contamination are important for coastal ecosystems, society, and the economy as evidenced by the Deepwater Horizon oil spill in the Gulf of Mexico in 2010 and the Fukushima nuclear plant incident in the Pacific Ocean in 2011. Accurate prediction of pollutant pathways and concentrations at the ocean surface requires understanding ocean dynamics over a broad range of spatial scales. Fundamental questions concerning the structure of the velocity field at the submesoscales (100 m to tens of kilometers, hours to days) remain unresolved due to a lack of synoptic measurements at these scales. Using high-frequency position data provided by the near-simultaneous release of hundreds of accurately tracked surface drifters, we study the structure of submesoscale surface velocity fluctuations in the Northern Gulf of Mexico. Observed two-point statistics confirm the accuracy of classic turbulence scaling laws at 200-m to 50-km scales and clearly indicate that dispersion at the submesoscales is local, driven predominantly by energetic submesoscale fluctuations. The results demonstrate the feasibility and utility of deploying large clusters of drifting instruments to provide synoptic observations of spatial variability of the ocean surface velocity field. Our findings allow quantification of the submesoscale-driven dispersion missing in current operational circulation models and satellite altimeter-derived velocity fields.