Extracting multiple unknown sources from a single observation of a single-channel is an ill-posed problem encountered in a variety of applications. This paper characterizes the ambiguity of solutions to the source separation problem, and then proposes a novel adaptive-operator-based approach to deriving solutions based on a combination of separation operators and domain-specific knowledge related to sources. The proposed scheme involves transforming the original problem into a new problem, in which data-dependent operators and the unknown sources are variables to be optimized. We demonstrate that a solution to the proposed optimization problem must reside in the null spaces of the operators, and any such solution also provides an optimal value to the original problem. We then demonstrate the applicability of the proposed method to the separation of sparse sources as well as AM-FM sources. Note that the proposed scheme outperformed corresponding state-of-the-art methods in noiseless as well as noisy environments. Finally, we demonstrate the efficacy of the proposed scheme in separation tasks based on real-world ECG data (i.e., extracting fetal ECG signals from noisy observations in which maternal and fetal ECGs recordings are superimposed) and electrical data (i.e.,separating singularities from harmonic components in an observation of noisy data related to surges in electrical current).