Cylindrical surface waves (CSWs) and slow-wave instabilities of a rectangularly corrugated cylinder are numerically examined. CSWs are slow waves with upper cutoffs at the point. The upper cutoff frequency increases with increasing cylindrical radius R0. There are two types of higher-order CSWs: one is due to azimuthal standing waves and the other is due to radial standing waves in the corrugation. Both higher-order types of SWSs have lower cutoffs as well as upper cutoffs leading to pass and stop bands. Slow space charge and slow cyclotron modes of an annular beam exist, which excite the Cherenkov and slow cyclotron instabilities of CSWs, respectively. The growth rates of the higher-order CSWs are comparable to those of the fundamental SWSs.