This paper considers the massive connectivity application in which a large number of potential devices communicate with a base-station (BS) in a sporadic fashion. The detection of device activity pattern together with the estimation of the channel are central problems in such a scenario. Due to the large number of potential devices in the network, the devices need to be assigned non-orthogonal signature sequences. The main objective of this paper is to show that by using random signature sequences and by exploiting sparsity in the user activity pattern, the joint user detection and channel estimation problem can be formulated as a compressed sensing single measurement vector (SMV) problem or multiple measurement vector (MMV) problem, depending on whether the BS has a single antenna or multiple antennas, and be efficiently solved using an approximate message passing (AMP) algorithm. This paper proposes an AMP algorithm design that exploits the statistics of the wireless channel and provides an analytical characterization of the probabilities of false alarm and missed detection by using the state evolution. We consider two cases depending on whether the large-scale component of the channel fading is known at the BS and design the minimum mean squared error (MMSE) denoiser for AMP according to the channel statistics. Simulation results demonstrate the substantial advantage of exploiting the statistical channel information in AMP design; however, knowing the large-scale fading component does not offer tangible benefits. For the multiple-antenna case, we employ two different AMP algorithms, namely the AMP with vector denoiser and the parallel AMP-MMV, and quantify the benefit of deploying multiple antennas at the BS.