Waiting on line at Radio Shack is a good example of a linked list. You get online at the back of the queue and get off at the front. There's no reason to access any elements in the middle. (there is even a Queue interface to make it obvious you are using a queue).
An array list is used as the "default collection type" even when another type could be used. I'm having trouble thinking of an example - probably because I just use it without thinking if another data type isn't needed - but anytime you need to know the index of an element is the use case.