The phase in quantum states is an essential information carrier for quantum telecommunications, signal processing, and computation. Quantum phase estimation is therefore a fundamental operation to extract and control useful information at the quantum level. Here, we analyze various approaches to quantum phase estimation, when a phase parameter characterizing a quantum process gets imprinted in a relative phase attached to a quantum state serving as a probe signal. The estimation approaches are based on standard concepts of signal processing (Fourier transform, maximum likelihood), yet operated in the quantum realm. We also exploit the Fisher information, both in its classical and its quantum forms, in order to assess the performance of each approach to quantum phase estimation. We demonstrate a possibility of enhanced estimation performance, inaccessible classically, which is obtained via optimized quantum entanglement. Beyond their significance to quantum phase estimation, the results illustrate how standard concepts of signal processing can contribute to the ongoing developments in quantum information and quantum technologies.