From a performance perspective losing an engine at V1 is most critical and will result in the most runway used and lowest obstacle clearing heights. From a pilot perspective a failure at rotation can be a bit more of a handful since the airplane in flight rolls and yaws as a result of the failure. In general with all aircraft using significant rudder to prevent the yaw combined with lowering the nose from a standard climb pitch to prevent decay of airspeed will result in a positive rate of climb and when at a safe altitude other procedures can be accomplished. Adding power on the good engine and in prop aircraft confirming feathering might also need to be done right after the failure but most jet aircraft have performance calculations for each takeoff that do not require pilots to add power to the good engine in the event of a failure.
I suffered my one and only engine failure at rotation with the nose wheel in the air but mains still on the ground. It was a contained failure of an interstage turbine on a turboprop but handling was fairly docile and I never felt in jeopardy. Our normal climb rate would have been 2500+ feet per minute but we managed just under 500 fpm as we climbed to pattern altitude and returned to the field. I currently fly the Boeing 737 and it would be even more docile with less pilot pilot workload and more performance should the same occur.