We investigate the problem of simultaneous propagation of multiple hydraulic fractures in situations when the effect of buoyancy is important. While such fractures often occur in nature as magmatic dikes, they also bear technological relevance. For instance, some petroleum reservoirs have strong upward stress gradient and multiple hydraulic fractures are often created simultaneously from a horizontal well to promote operational efficiency. More recently, enhanced geothermal systems employed a similar multi-fracture from a horizontal well approach to stimulate more homogeneous hot rock. The stress state in the latter rock formations is often dominated by a constant stress gradient and promotes buoyancy effects. In geothermal applications, there is at least one more additional well that is drilled to establish circulation. Connectivity between the wells is therefore a crucial aspect of the treatment design. For this reason, we aim to investigate hydraulic fracture morphology of multiple fractures, since it is directly related to connectivity between the wells. To address the problem, we utilize a recently constructed parametric space for a single buoyancy-driven hydraulic fracture, and analyze the fracture shapes for the multi-fracture cases in this dimensionless space to ensure coverage of all possible scenarios. We find that hydraulic fractures exhibit instabilities in all of the cases by trying to prevent an overlap between the fractures. As a result, fractures form multiple fracture “heads” or fingers. At the same time, the results noticeably depend on the fracture propagation regime respectively on the location in the parametric space. In particular, horizontal fracture extension, vertical fracture height, aperture, and the number of “heads” vary considerably within the parametric space. Implications of the results to practical applications are discussed and recommendations on instability avoidance are provided.