The stability of the two-dimensional flow induced by the transverse oscillation of a cylinder in a viscous fluid is investigated in both the linear and weakly nonlinear regimes. The major assumption that is made to simplify the problem is that the oscillation frequency is large, in which case an unsteady boundary layer is set up on the cylinder. The basic flow induced by the motion of the cylinder depends on two spatial variables, and is periodic in time. The stability analysis of this flow to axially periodic disturbances therefore leads to a partial differential system dependent on three variables. In the high-frequency limit the linear stability problem can be reduced to a system dependent only on a radial variable and time. Furthermore, the coefficients of the differential operators in this system are periodic in time, so that Floquet theory can be used to reduce this system further to a coupled infinite system of ordinary differential equations together with uncoupled homogeneous boundary conditions. The eigenvalues of this system are found numerically and predict instability entirely consistent with the experiments with circular cylinders performed by Honji (1981). Results are given for cylinders of elliptic cross-section, and it is found that for any given eccentricity the most dangerous configuration is when the cylinder oscillates parallel to its minor axis. Some discussion of nonlinear effects is also given, and for the circular cylinder it is shown that the steady-streaming boundary layer of the basic flow is significantly altered by the instability.