Sit down here and forget your troubles (OR problems) for a minute.
I thought that the word “trouble” had been always considered as
an uncountable noun untill I found those sentences noted above.
How should I use those countable “troubles”?
In what situation are we allowed to use them?
The countable use of “troubles” is very common. The difference is whether you perceive it as a mere state of difficulty, which is the uncountable one, or one or several examples of difficulty.
We have overcome our financial troubles.
We think we have found a solution to troubles we have experienced with new electronic voting machines.
Don’t let your troubles get you down.
The poor man has more troubles than you could count.
Try substituting the two synonyms you mention – ‘problems’ or ‘worries’ – where you want to use ‘trouble/troubles’.
If within the context of your sentence the word ‘problem’ or ‘worry’ is singular, then use ‘trouble’. If the word you require is the plural ‘problems’ or ‘worries’, then use ‘troubles’.