If the target permanent is an illegal target by the time Assassin's Trophy tries to resolve, the spell doesn't resolve. No player searches their library.

If the target permanent is a legal target but isn't destroyed, most likely because it has indestructible, its controller may search their library.

If the permanent's controller doesn't search their library, they don't shuffle their library.

If the target permanent is an illegal target by the time Assassin's Trophy tries to resolve, the spell doesn't resolve. No player searches their library.

If the target permanent is a legal target but isn't destroyed, most likely because it has indestructible, its controller may search their library.

If the permanent's controller doesn't search their library, they don't shuffle their library.