Context: The location of pure frequencies in the spectrum of an irregularly sampled time series is an important topic in astrophysical data analysis. Especially in the domain of asteroseismology, a highly precise and unambiguous study of frequencies in photometric light or radial velocity curves is required. Aims: Due to sampling irregularities and large observational gaps, the classic methods for frequency estimation (prewhitening techniques, clean, cleanest, etc.) sometimes suffer false detections. We propose a new framework for this problem that allows a more precise and unambiguous frequency location. Methods: Multisine fitting is addressed as the sparse representation of the data in an overcomplete dictionary of frequencies, hence the name SparSpec for the method. We model the data as the sum of an arbitrarily large number of pure frequencies, discretised on a fixed grid. Among all the many representations fitting the data, we seek the one with the fewest non-zero amplitudes. This solution can be computed by minimising a convex criterion with no local minima. A computationally efficient and convergent optimisation strategy is derived and a user-friendly software implementing SparSpec is provided online at http://www.ast.obs-mip.fr/Softwares. Results: The method is first illustrated on a simple test example where SparSpec correctly locates the frequencies while classic methods fail. Then, simulations on more realistic artificial time series reveal the interest of this new methodology in terms of robustness toward sampling aliases. An application to the radial velocity curve of the pre-main sequence Herbig Ae star HD 104237 is finally presented, where the method is able to determine oscillation frequencies even in the presence of strong low-frequency perturbations such as orbital movements. While SparSpec mainly confirms previously published studies for the four more important frequencies, it suggests some ambiguity about the position of a fifth frequency. Additional simulations show that the SparSpec results are more plausible.