An important problem in oil spill containment by booms is the instability of the oil-water interface at the boom. This instability, which represents the conditions under which oil can escape under the boom, is investigated. A viscous flow model for thin slicks in two dimensions is developed. To understand the effect of viscosity on the instability criterion, the full Navier-Stokes equations are solved by the fractional-step method in time-domain to determine the pressure gradients along the boom. The numerically obtained viscous instability criterion Analytical instability formulas for potential flows are based on the velocity potentials for attached and detached flows due to uniform current past a flat plate in finite and infinite water depths. The results show that the viscous flow model predicts a larger region of stability. It is numerically determined from the instability criterion that the oil droplets at the boom between the free surface and down to about 40 percent of the boom height can never escape, regardless of the current strength. It is also shown that the instability criterion depends weakly on the high Reynolds number. Reanalysis of the available experimental data confirms these findings.