We describe two-dimensional models with a metallic Fermi surface which display quantum phase transitions controlled by strongly interacting critical field theories below their upper critical dimension. The primary examples involve transitions with a topological order parameter associated with dislocations in collinear spin-density-wave ("stripe") correlations: the suppression of dislocations leads to a fractionalization of spin and charge collective modes, and this transition has been proposed as a candidate for the cuprates near optimal doping. The coupling between the order parameter and long-wavelength volume and shape deformations of the Fermi surface is analyzed by the renormalization group, and a runaway flow to a nonperturbative regime is found in most cases. A phenomenological scaling analysis of simple observable properties of possible second-order quantum critical points is presented, with results quite similar to those near quantum spin-glass transitions and to phenomenological forms proposed by Schroder [Nature (London) 407, 351 (2000)].