We study a sampling problem where sampled signals come from a known union of shift-invariant subspaces and the sampling operator is a linear projection of the sampled signals into a fixed shift-invariant subspace. In practice, the sampling operator can be easily implemented by a multichannel uniform sampling procedure. We present necessary and sufficient conditions for invertible and stable sampling operators in this framework, and provide the corresponding minimum sampling rate. As an application of the proposed general sampling framework, we study the specific problem of spectrum-blind sampling of multiband signals. We extend the previous results of Bresler et al. by showing that a large class of sampling kernels can be used in this sampling problem, all of which lead to stable sampling at the minimum sampling rate.