Difficulty or difficulties falling asleep
There is no question of grammar here. Both are completely grammatical.
However, one is idiomatic and the other is not.
You can use "had difficulty" whenever somebody tried to do something and found it hard - even if they encountered more than one problem.
If you say they "had difficulties", you are putting emphasis on the fact that there were several successive problems. It is likely that they were trying to do an action that required several separate steps.