We experimentally investigate the interfacial instabilities governing the dynamics of an interface between two superposed immiscible liquids (oil and water) in a cylindrical container oscillating about its axis. The viscosity and density contrasts are $100$ and $0.968$, respectively. Depending on the vibrational Froude number, the evolution of interfacial wave is categorized into single-droplet (SD) formation (at the core region) and multiple/emulsion-droplet formation (at the near-wall region), and the breakage of the deformed interface into a SD is analysed for the first time. The thresholds for the onset of different instabilities responsible for each regime are presented by the amplitude and frequency of rotation, of which the boundaries predicted through the inviscid theory and scaling arguments are in good agreement with measurement. For SD formation, in particular, it is related to the critical rise velocity of the interface, represented by the vibrational Froude number. We emphasize the opposing contributions between (i) the viscous effect, i.e. the dimensionless thickness of the Stokes boundary layer, and (ii) the inviscid effect, i.e. the dimensionless maximum interface rise at the centre region (inviscid core), promoting and preventing the formation of a falling jet, respectively, which is necessary for SD formation. Our results indicate that viscosity plays an important role in shaping the boundary of SD and multiple-droplet regimes, especially at a relatively small (high) oscillating amplitude (frequency). When the amplitude is small, the enhanced viscous effect forces the deformed interface to migrate to multiple-droplet formation, skipping SD formation, with increasing frequency.