We construct general models for holographic superconductivity parametrized by three couplings which are functions of a real scalar field and show that under general assumptions they describe superconducting phase transitions. While some features are universal and model independent, important aspects of the quantum critical behavior strongly depend on the choice of couplings, such as the order of the phase transition and critical exponents of second-order phase transitions. In particular, we study a one-parameter model where the phase transition changes from second to first order above some critical value of the parameter and a model with tunable critical exponents.