We propose a maximum likelihood estimation approach for the recovery of continuously-defined sparse signals from noisy measurements, in particular periodic sequences of Diracs, derivatives of Diracs and piecewise polynomials. The conventional approach for this problem is based on least-squares (a.k.a. annihilating filter method) and Cadzow denoising. It requires more measurements than the number of unknown parameters and mistakenly splits the derivatives of Diracs into several Diracs at different positions. Moreover, Cadzow denoising does not guarantee any optimality. The proposed approach based on maximum likelihood estimation solves all of these problems. Since the corresponding log-likelihood function is non-convex, we exploit the stochastic method called particle swarm optimization (PSO) to find the global solution. Simulation results confirm the effectiveness of the proposed approach, for a reasonable computational cost.