During takeoff, which speeds are used to assess rotation and engine-out performance?

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The key idea is matching speeds to the takeoff actions: when to rotate, and what speed means you can safely continue with an engine out. The rotation speed is VR, the speed at which you lift the nose to get airborne. For engine-out performance, you need to be able to maintain a safe climb with one engine inoperative, which is defined by V2, the takeoff safety speed. In addition, V1 is the decision speed: if an engine failure occurs before V1 you should abort the takeoff; if it happens at or after V1 you continue. So the speeds used to assess rotation and engine-out performance are VR for rotation, V2 for takeoff performance with an engine out, and V1 as required by the takeoff profile. Other options don’t match these roles: Vne/Vmo are maximum operating speeds, and Vs1 is the stall speed in takeoff configuration, not the rotation or engine-out performance speeds.