We use a number of large-N limits to explore the competition between ground states of square lattice doped antiferromagnets which break electromagnetic U(1), time-reversal, or square lattice space group symmetries. Among the stable states we find are d-, (s+id)-, and (d+id)-wave superconductors, Wigner crystals, Wigner crystals of hole pairs, orbital antiferromagnets (or staggered-flux states), and states with spin-Peierls and bond-centered charge stripe order. In the vicinity of second-order quantum phase transitions between the states, we go beyond the large-N limit by identifying the universal quantum field theories for the critical points, and computing the finite temperature, quantum-critical damping of fermion spectral functions. We identify candidate critical points for the recently observed quantum-critical behavior in photoemission experiments on BSCCO by Valla et al. (Science 285, 2110 (1999)): these involve onset of a charge density wave in a d-wave superconductor, or broken time-reversal symmetry with (d+id) or (s+id) pairing, and lead to the observed fermion damping in the vicinity of the nodal points in the Brillouin zone. The latter cases, with broken time-reversal symmetry, are appealing because the order parameter is not required to satisfy any special commensurability conditions. The observed absence of inelastic damping of quasiparticles with momenta (pi,k), (k,pi) (with 0 < k < pi) also appears very naturally for the case of a transition to (d+id) order.