The engine misfire detection process according to one aspect of the invention first makes tentative detection on the occurrence or the non-occurrence of an engine misfire in execution of both vibration control and rotation speed control. In response to the tentative detection of an engine misfire, the engine misfire detection process subsequently makes final detection on the occurrence or the non-occurrence of an engine misfire. In the event of no final detection of an engine misfire, the engine misfire detection process makes tentative detection and final detection on the occurrence or the non-occurrence of an engine misfire in prohibition of the vibration control. In the event of still no final detection of an engine misfire, the engine misfire detection process makes tentative detection and final detection on the occurrence or the non-occurrence of an engine misfire in further prohibition of the rotation speed control. Even in the event of no final detection of an engine misfire in execution of both the vibration control and the rotation speed control, the procedure of the invention makes final detection on the occurrence or the non-occurrence of an engine misfire in prohibition of execution of the vibration control and further the rotation speed control. This arrangement desirably increases the detection frequency of engine misfires.