We provide numerical and analytic evidence for the formation of a singularity driven only by surface tension in the mathematical model describing a two‐dimensional Hele–Shaw cell with no air injection. Constantin and Pugh have proved that no such singularity is possible if the initial shape is close to a circle; thus we show that their result is not true in general. Our evidence takes the form of direct numerical simulation of the full problem, including a careful assessment of the effects of limited spatial resolution, and comparison of the full problem with the lubrication approximation. ©1996 American Institute of Physics.