Summary: The linear stability of a periodic array of vortices in stratified fluid is studied by modal stability analysis. Two base flows are considered: the two-dimensional Taylor-Green vortices and the Stuart vortices. In the case of the two-dimensional Taylor-Green vortices, four types of instability are identified: the elliptic instability, the pure hyperbolic instability, the strato-hyperbolic instability and the mixed hyperbolic instability, which is a mixture of the pure hyperbolic and the strato-hyperbolic instabilities. Although the pure hyperbolic instability is most unstable for the non-stratified case, it is surpassed by the strato-hyperbolic instability and the mixed hyperbolic instability for the stratified case. The strato-hyperbolic instability is dominant at large wavenumbers. Its growth rate tends to a constant along each branch in the large-wavenumber and inviscid limit, implying that the strato-hyperbolic instability is not stabilized by strong stratification. Good agreement between the structure of the strato-hyperbolic instability mode and the corresponding local solution is observed. In the case of the Stuart vortices, the unstable modes are classified into three types: the pure hyperbolic instability, the elliptic instability and the mixed-type instability, which is a mixture of the pure hyperbolic and the elliptic instabilities. Stratification decreases the growth rate of the elliptic instability, which is expected to be stabilized by stronger stratification, although it is not completely stabilized within the range of Froude numbers considered. The present results imply that both the pure hyperbolic instability and the strato-hyperbolic instability are important in stably stratified flows of geophysical or planetary scale.