Array processing algorithms are used in many applications for source localization and signal waveform estimation. When the number of snapshots is small and/or the signal-to-noise ratio (SNR) is low, it becomes a challenge to discriminate closely-spaced sources. In this paper, two new array processing algorithms exploiting sparsity are proposed to overcome this problem. The first proposed method combines a well-known sparsity preserving algorithm, namely the least absolute shrinkage and selection operator (LASSO), with the Bayesian information criterion (BIC) to eliminate user parameters. The second proposed algorithm extends the sphere decoding algorithm, which is widely used in communication applications for the recovery of signals belonging to a finite integer dictionary, to promote the sparsity of the solution. The proposed algorithms are compared with several existing sparse signal estimation techniques. Simulations involving uncorrelated and coherent sources demonstrate that the proposed algorithms, especially the algorithm based on sphere decoding, show better performance than the existing methods. Moreover, the proposed algorithms are shown to be more practical than the existing methods due to the easiness in selecting their user parameters.