Analyzing data mapped to the sphere as may occur in a range of applications in geophysics, medical imaging or astrophysics, requires specific tools. This paper describes new multiscale decompositions for spherical images namely the isotropic undecimated wavelet transform, the ridgelet transform and the curvelet transform each of which is invertible. Several applications are described. We show how these transforms can be used in denoising and especially in a Combined Filtering Method, which uses both the wavelet and the curvelet transforms, thus benefiting from the advantages of both transforms. An application to component separation from multichannel data mapped to the sphere is also described where we take advantage of the spatiospectral localization on the sphere provided by the spherical wavelet functions.