Surface tension can cause a deformed liquid drop to break into several droplets. A fluid interface subjected to external stresses spontaneously forms sharp points and thin, extended filaments and sheets. We examine two examples of singularity formation on the fluid interface: the breakup of a water drop in oil and the steady-state entrainment of water by oil. We show that the singularity-formation dynamics in these two systems are very different from previously studied situations, in which the dynamics near a surface singularity becomes scale-invariant and independent of large-scale conditions. In both examples here, conditions on the largest length- and time-scales have a significant effect on the dynamics near the singularity without destroying scaling behavior.