We present a Bayesian approach for nonparametric curve estimation based on a continuous wavelet dictionary, where the unknown function is modeled by a random sum of wavelet functions at arbitrary locations and scales. By avoiding the dyadic constraints for orthonormal wavelet bases, the continuous overcomplete wavelet dictionary has greater flexibility to adapt to the structure of the data, and leads to sparse representations. The price for this flexibility is the computational challenge of searching over an infinite number of potential dictionary elements. We develop a reversible jump Markov Chain Monte Carlo algorithm which uti-lizes local features in the proposal distributions and leads to better mixing of the Markov chain. Performance comparison in terms of sparsity and mean square error is carried out on standard wavelet test functions. Results on a non-equally spaced example show that our method compares favorably to methods using interpolation or imputation.