Yes they were a rage in C for a while and there are of course advantages to using a linked list, like simulating a stack or queue, inserting and deleting items is faster. Also, it's easier to use a neighbouring object directly from another object in iterations. The downside is the overall performance and accessing items in the list. Arraylists are much faster in general and you can access the nth item quickly (myArray[n]). In a LinkedList you have iterate over all items to get to the nth one. And last but not least, sorting linked lists is difficult.
What I don't understand is that Microsoft decided to give us a LinkedList collection in the .NET Framework 2.0 (it's a doubly linked list, so each node points forward and backwards).
The only useful application for a linked list I can think of is something like a preview. For example, you are displaying information on the screen and the user can click buttons to go to the next or previous item. You could easily display information about the next and/or previous items using a linked list; this would be more cumbersome in a normal list I guess.