Besides improving communication performance, intelligent reflecting surfaces (IRSs) are also promising enablers for achieving larger sensing coverage and enhanced sensing quality. Nevertheless, in the absence of a direct path between the base station (BS) and the targets, multi-target sensing is generally very difficult, since IRSs are incapable of proactively transmitting sensing beams or analyzing target information. Moreover, the echoes of different targets reflected via the IRS-assisted virtual links arrive at the BS from the same direction. In this paper, we study a wireless system comprising a multi-antenna BS and an IRS for multi-target sensing, where the beamforming vector and the IRS phase shifts are jointly optimized to improve the sensing performance. To meet the different sensing requirements, such as a minimum received power and a minimum sensing frequency, we propose three novel IRS-assisted sensing schemes: Time division (TD) sensing, signature sequence (SS) sensing, and hybrid TD-SS sensing. For TD sensing, the sensing tasks are performed in sequence over time. In contrast, the novel SS sensing scheme senses all targets simultaneously and establishes a relationship between the target directions and SSs. To strike a flexible balance between the beam pattern gain and sensing efficiency, we also propose a general hybrid TD-SS sensing scheme with target grouping, where targets belonging to the same group are sensed simultaneously via SS sensing, while the targets in different groups are assigned to orthogonal time slots. By controlling the number of groups, hybrid TD-SS sensing can provide a more flexible balance between beam pattern gain and sensing frequency. Moreover, we propose a two-layer penalty-based algorithm to solve the challenging non-convex optimization problem for the joint design of the BS beamformers, IRS phase shifts, and target grouping. Simulation results demonstrate the effectiveness of the proposed hybrid scheme in achieving a flexible trade-off between beam pattern gain and sensing frequency. Our results also reveal that the power leakage in unintended directions is larger for tighter interference constraints.