The interaction between fluid and thin plates, accompanied with flow separation, vortex shedding and wake transition, is of great significance in engineering applications and fluid mechanics. Some key parameters, i.e. the Reynolds number (Re), the incidence angle (α) and the planforms of plates controlled by edge number (n) and shrinkage ratio (k), determine the flow patterns of thin plates. Extensive direct numerical simulations are performed to study the effects of these parameters on the flow around thin plates. In the considered parametric spaces (Re – k & n and Re - α), several flow states and their distribution are obtained, i.e. SS, RSB, RSP, QP and CS for vertically fixed plates, and SS, PS, PSL, QP and CS for inclined plates. Typical features and critical boundaries between different flow regimes are reported and discussed in brief. The comprehensive parametric studies give us a deeper understanding of dominant role of Re in the wake transition behind thin plates from the steady to unsteady state and eventually the chaotic state. The wake transition is quite similar regardless of thin plate planforms, but affect the critical Reynolds number. For inclined plates, α plays a role in modifying the wake patterns, e.g. selecting the plane of reflectional symmetry, changing the frequency of primary vortex shedding and controlling the vortex shedding mechanism. Additionally, we also conduct studies on the flow induced by a circular disk oscillating sinusoidally along its axis. In the parametric space (50 ≤ Re ≤ 800 & 1 ≤ KC ≤ 24), five flow regimes are observed: AS, PSL, ARL, PSH and ARH. For the non-axisymmetric flow regimes, the flow is one-sided with respect to the axis of the disk and is associated with a non-zero mean value of the transverse force acting on the disk.