We present in this paper a-posteriori error estimators for multiphase flow with singular well sources. The estimators are fully and locally computable, distinguish the various error components, and target the singular effects of wells. On the basis of these estimators we design an adaptive fully-implicit solver that yields optimal nonlinear iterations and efficient time-stepping, while maintaining the accuracy of the solution. A key point is that the singular nature of the solution in the near-well region is explicitly captured and efficiently estimated using the adequate norms. Numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive algorithm.