Which sentence is correct between those two:
The problems about the waste-sorting are ...
The problems with waste-sorting are ...
The problems about the waste-sorting are ... The problems with waste-sorting are...
Both sentences are correct.
"The problem(s) with ..." is a common noun phrase (as part of a phrase). "The problem(s) about ..." is similar noun phrase but "with" is used more often (as far as I've experienced) and can sound more natural.