A body travelling towards a stationary finite body collides with...

Let the mass of the moving body be $m_1$ with initial velocity $u_1 \neq 0$.

Let the mass of the stationary finite body be $m_2$ with initial velocity $u_2 = 0$.

The initial momentum of the system is $P_i = m_1 u_1 \neq 0$.

According to the law of conservation of linear momentum, the final momentum $P_f$ must be equal to $P_i$.

If both bodies come to rest, $P_f = 0$, which violates the conservation of momentum. Thus, the first outcome is not possible.

If the stationary body remains at rest, its change in momentum is zero, meaning no impulse acted on it. By Newton's third law, no impulse acted on the moving body either, so its velocity cannot change. Thus, the fourth outcome is not possible.

If the collision is perfectly elastic and $m_1 = m_2$, the bodies exchange velocities. The moving body comes to rest and the stationary body moves with velocity $u_1$. Thus, the third outcome is possible.

For other mass ratios or inelastic collisions, both bodies can be in motion after the collision. Thus, the second outcome is possible.