Reconstructing images from very long baseline interferometry (VLBI) data with sparse sampling of the Fourier domain (uv-coverage) constitutes an ill-posed deconvolution problem. It requires application of robust algorithms maximizing the information extraction from all of the sampled spatial scales and minimizing the influence of the unsampled scales on image quality. We present novel multiscale wavelet deconvolution algorithms for imaging sparsely sampled interferometric data. These new ideas are based on a novel, specially designed wavelet dictionary and hard image thresholding in the spirit of compressive sensing. Compressing various spatial features of the true sky brightness distribution by various scales provides a powerful way to analyse the uv-coverage during imaging and improving the seperation between covered features and features introduced by gaps in the uv-coverage. We demonstrate the stability of our novel algorithmic ideas and benchmark their performance against image reconstructions made with CLEAN and Regularized Maximum-Likelihood (RML) methods using synthetic data. The comparison shows that multiscalar approaches match the superresolution achieved by the RML reconstructions and surpass the sensitivity to extended emission reached by CLEAN. Moreover, the imaging is largely data-driven reducing the human induced bias during the imaging procedure. Finally, we present some natural extensions to dynamic imaging, polarimetry and finally dynamic polarimetry