Which of the two can be zero under certain conditions: average speed or average velocity of a moving body?

The average speed of a moving body cannot be 0. It can be possible only when the body is at rest.
While the average velocity of a moving body can be zero provided it should be moving in a circular path. Displacement is the shortest path between the initial and the final position. When the initial and the final positions are the same then displacement becomes zero making velocity zero.
Example: When we run in a circular path we return to the same/initial position, so our displacement will be zero so as our velocity.