While there has been increasing studies of MoS2 and other two-dimensional (2D) semiconducting dichalcogenides on hard conventional substrates, experimental or analytical studies on flexible substrates has been very limited so far, even though these 2D crystals are understood to have greater prospects for flexible smart systems. In this article, we report detailed studies of MoS2 transistors on industrial plastic sheets. Transistor characteristics afford more than 100x improvement in the ON/OFF current ratio and 4x enhancement in mobility compared to previous flexible MoS2 devices. Mechanical studies reveal robust electronic properties down to a bending radius of 1 mm which is comparable to previous reports for flexible graphene transistors. Experimental investigation identifies that crack formation in the dielectric is the responsible failure mechanism demonstrating that the mechanical properties of the dielectric layer is critical for realizing flexible electronics that can accommodate high strain. Our uniaxial tensile tests have revealed that atomic-layer-deposited HfO2 and Al2O3 films have very similar crack onset strain. However, crack propagation is slower in HfO2 dielectric compared to Al2O3 dielectric, suggesting a subcritical fracture mechanism in the thin oxide films. Rigorous mechanics modeling provides guidance for achieving flexible MoS2 transistors that are reliable at sub-mm bending radius.