The instability of a solitary flexible cylinder in three-dimensional axial flow with various boundary conditions, lengths, and Reynolds numbers has been investigated by CFD (computational fluid dynamics) simulation. A new CFD code is modified from an existing code for a two-dimensional FSI (fluid-structure interaction) problem. The fluid field is solved by the BDIM (boundary data immersion boundary method). The solid structure's displacement is solved by the IGA (isogeometric analysis). The coupling effect is solved by the partitioned and implicit method combined with the IQN (interface Quasi-Newton) method. The simulated results of a clamped-free cylinder can match the experimental results quantitatively. The investigation for cylinders with different parameters indicates that the instability transition as cylinder length increases is similar to that of a clamped-free cylinder as flow velocity increases, small vibrations around the non-zero neutral position are observed when cylinder is in divergence, the deflection amplitude during the transition range reduces, an extra blunt downstream end can reduce the flutter amplitude, and increasing the fluid viscosity in the viscous force exemption range has no stabilizing effect.