We present a new method for the selection of individual sperm cells using a microfluidic device that automatically traps each cell in a separate microdroplet that then individually self-assembles with other microdroplets, permitting the controlled measurement of the cells using quantitative phase microscopy. Following cell trapping and droplet formation, we utilize quantitative phase microscopy integrated with bright-field imaging for individual sperm morphology and motility inspection. We then perform individual sperm selection using a single-cell micromanipulator, which is enhanced by the microdroplet-trapping procedure described above. This method can improve sperm selection for intracytoplasmic sperm injection, a common type of in vitro fertilization procedure.