Recently, intelligent reflecting surface (IRS)-assisted communication has gained considerable attention due to its advantage in extending the coverage and compensating the path loss with low-cost passive metasurface. This paper considers the uplink channel estimation for IRS-aided multiuser massive MISO communications with one-bit ADCs at the base station (BS). The use of one-bit ADC is impelled by the low-cost and power efficient implementation of massive antennas techniques. However, the passiveness of IRS and the lack of signal level information after one-bit quantization make the IRS channel estimation challenging. To tackle this problem, we exploit the structured sparsity of the user-IRS-BS cascaded channels and develop three channel estimators, each of which utilizes the structured sparsity at different levels. Specifically, the first estimator exploits the elementwise sparsity of the cascaded channel and employs the sparse Bayesian learning (SBL) to infer the channel responses via the type-II maximum likelihood (ML) estimation. However, due to the one-bit quantization, the type-II ML in general is intractable. As such, a variational expectation-maximization (EM) algorithm is custom-derived to iteratively compute an ML solution. The second estimator utilizes the common row-structured sparsity induced by the IRS-to-BS channel shared among the users, and develops another type-II ML solution via the block SBL (BSBL) and the variational EM. To further improve the performance of BSBL, a third two-stage estimator is proposed, which can utilize both the common row-structured sparsity and the column-structured sparsity arising from the limited scattering around the users. Simulation results show that the more diverse structured sparsity is exploited, the better estimation performance is achieved, and that the proposed estimators are superior to state-of-the-art one-bit estimators.