First example of inertia of rest: Fall of dust particles from a durries or cloth: We all have seen that the dust particles in a durries or cloth fall off when it is beaten with a stick. This is due to the reason that the beating results in motion of durries but the dust particles remain at rest. Thus dust particles fall off. This is one of the examples of inertia of rest.

The passenger in a running bus tends to lean forward , when the bus stops suddenly:

When the bus is running, the whole body of the passenger is in the state of motion. When the bus stops suddenly, the lower part of the passengers body which is in contact with the bus, comes to rest. But the passengers upper portion remains in the state of motion due to the inertia of motion. This is the reason for the passenger to lean forward when the bus stops suddenly.
The passengers sitting in any vehicle lean forward when suddenly a vehicle is stopped due to inertia, the dust particles come out of a cloth when beaten due to inertia, etc.....