I hate the ones where the only way to proceed is to guess and then see what happens.
Those don't exist.
When I've reached a point where I can't find a single definitive spot, I move on to another puzzle. Sometimes a few days provides a fresh set of eyes that will see something you missed.
That can be hard when you use pen notes (on paper puzzles) in the corners. When you look at it after a long time, it's even worse looking at the puzzle.
Most Medium and some Hard puzzles I won't use notes, or may at most use one or two notes early for a few squares.
Most Super Hard puzzles I'll have nearly the whole puzzle scattered with notes.
But the short answer to the OP:
There is one solution to the puzzle - no square will have more than one correct number. If a square appears that way at one point in time, it's because you have not discovered enough solutions to deduce that specific box just yet. Myriad combinations across the whole puzzle can unlock any given box.
Sometimes I seem to find a wall where I can find no further solutions; Sometimes I literally have to give up, especially when I'm feeling foggy headed and attempting some of the harder puzzles out there.
One of the few pleasures of my shitty (unrelated) job. I can put hours into puzzles (or reading, or whatever is not delivered by an electronic device ) and sometimes only have less than 10 minutes of interruptions in an hour.
Sometimes I have put more than two hours of time into a single puzzle before I solved it. I'm stubborn and hate quitting, so I have to have spent a good time staring at it, stabbing for answers, before I give up. Sometimes that persistence pays off and I find a key answer that goes on to make a whole ton jump out at me.