The classical problem of nonlinear capillary waves on two-dimensional fluid sheets is reconsidered. The problem is formulated in terms of a complex potential, and solutions are sought using Fourier series expansions. A collocation technique combined with Newton's method is used to compute the Fourier coefficients numerically. Using this procedure, the exact solutions of Kinnersley (1976) are recomputed and various symmetric and antisymmetric wave profiles are presented, including the limiting configurations which exhibit trapped bubbles of air. Most important, three new solution branches which bifurcate nonlinearly from the symmetric Kinnersley solution branch are identified. The wave profiles along these new branches do not possess the symmetry or antisymmetry of the Kinnersley solutions, although their limiting configurations also display trapped air bubbles. No bifurcations are found along the antisymmetric Kinnersley solution branch.