Spectroscopy---analyzing the frequency spectrum of a time-dependent signal---is one of the most ubiquitous and versatile problems in natural science. An important question is to understand how the advances of quantum technologies can improve spectroscopy. Here, we develop a framework for spectroscopy via quantum sensing by studying signals supported on a known set of $N$ candidate frequencies. This discrete spectral model captures a broad range of settings with prior spectral information. We formulate three basic sensing tasks: signal detection, signal search, and signal learning. We develop protocols that solve each of the tasks and rigorously establish their optimality with respect to both $N$ and the target precision $\epsilon$ in each problem. Furthermore, we show that these tasks require different quantum resources: detection and search are solved optimally with a simple protocol in which sensor qubits undergo pulsed control sequences, similar to many exiting quantum sensing protocols. In contrast, the optimal signal learning requires joint coherent processing between the sensor qubits and ancilla qubits. Together, these results identify the fundamental limits of spectroscopy via quantum sensing and the quantum resources needed to achieve them.

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