In this paper, we investigate an intelligent reflecting surface (IRS)-assisted integrated sensing and communication (ISAC) system design in a clutter environment. Assisted by an IRS equipped with a uniform linear array (ULA), a multi-antenna base station (BS) is targeted for communicating with multiple communication users (CUs) and sensing multiple targets simultaneously. We consider the IRS-assisted ISAC design in the case with Type-I or Type-II CUs, where each Type-I and Type-II CU can and cannot cancel the interference from sensing signals, respectively. In particular, we aim to maximize the minimum sensing beampattern gain among multiple targets, by jointly optimizing the BS transmit beamforming vectors and the IRS phase shifting matrix, subject to the signal-to-interference-plus-noise ratio (SINR) constraint for each Type-I/Type-II CU, the interference power constraint per clutter, the transmission power constraint at the BS, and the cross-correlation pattern constraint. Due to the coupling of the BS's transmit design variables and the IRS's phase shifting matrix, the formulated max-min IRS-assisted ISAC design problem in the case with Type-I/Type-II CUs is highly non-convex. As such, we propose an efficient algorithm based on the alternating-optimization and semi-definite relaxation (SDR) techniques. In the case with Type-I CUs, we show that the dedicated sensing signal at the BS is always beneficial to improve the sensing performance. By contrast, the dedicated sensing signal at the BS is not required in the case with Type-II CUs. Numerical results are provided to show that the proposed IRS-assisted ISAC design schemes achieve a significant gain over the existing benchmark schemes.