A typical dielectric-barrier-discharge plasma actuator operating in burst mode generates periodic vortices resembling the starting vortex. This paper presents the three-dimensional (3-D) characteristics and instability mechanism of these vortices. The experimental investigation is carried out using smoke visualisation and time-resolved particle image velocimetry techniques in three orthogonal measurement planes. The size of the vortices decreases with an increase in burst signal frequency, $ f_{b}$, at a constant duty cycle, $ \alpha$. At higher burst frequencies, dipole vortices are formed due to the roll-up of the wall boundary layer. The angle of travel also decreases with an increase in $ f_{b}$. The evolution of $ \lambda _{2}$-criterion clearly demonstrates the vortex merging of co-rotating vortices. The vortex merging occurs at a critical ratio $ a_{c}/l_{c}$ of core size, $ a_{c}$, and separation distance, $ l_{c}$, equal to $ 0.22\pm 0.01$ which is close to $ a_{c}/l_{c} = 0.24\pm 0.01$ reported by Meunier et al. (Phys. Fluids,vol.14, 2002, pp. 2757–2766) for merging of a pair of equal two-dimensional co-rotating vortices. The periodic vortices are self-similar in nature and the vorticity distribution inside their core region follows the Lamb–Oseen vortex model. Cell structures form in the spanwise direction, which develops wave-like behaviour with an increase in burst frequency. Subsequently, these cell-like structures separate from each other, whose size and spacing correlate well with that of vorticity patches. The alternating sign of vorticity indicates that the circular cells have rotational motion in opposite sense with respect to each other. These cells grow downstream and appear in pairs of counter-rotating vortices (vortex dipole) akin to mushroom-like structures. At low values of $ \alpha$ and $ f_{b}$, the periodic vortex is subjected to a very weak strain and centrifugal instability dominates. The vortices are subjected to a higher strain at elevated burst frequencies, leading to the elliptic instability phenomenon similar to that observed in counter-rotating (Leweke & Williamson, J. Fluid Mech. 1998, vol. 360, pp. 85–119) and co-rotating (Meunier & Leweke, J. Fluid Mech.2005, vol. 533, pp. 125–159) vortex pair generated in water. The present experimental results based on the cross-cut visualisation, Galilean streamlines and vorticity decomposition confirm the role of the instability mechanism on the 3-D vortical structures generated by the dielectric-barrier-discharge plasma actuator.