In submarine geological and hydrodynamic environments, either tunnel erosion or lateral instability could be initiated where there is a shallowly embedded pipeline. Unlike previous studies on the tunnel erosion of sand and the lateral instability of pipelines, in this study we performed correlation analyses on the competition mechanism for these two physical processes. By correlating the critical flow velocities of these two processes, the instability envelope for the pipe-soil interaction system is established, which can be described using three key parameters: the embedment-to-diameter ratio, the dimensionless submerged weight of the pipe, and the corresponding critical flow velocity. The analysis procedure is further proposed to first determine the instability mechanism and then the critical velocity of ocean currents. Our parametric study indicates that tunnel erosion is more prone to emerging than lateral instability with small embedment-to-diameter ratio values. With increasing pipeline embedment, tunnel erosion can be suppressed and lateral instability therefore occurs more frequently. Moreover, for light pipelines, lateral instability is more likely to be triggered than tunnel erosion.