There's a large question with your question, a question that needs to be answered before we can answer "Which is the best jungle item?". That question is: who is on the enemy team and who is on your team?
Do you see the problem? The thing about LoL is that there are a lot of things that have a clear answer, e.g. "Which do I max first on Gnar?" but the far, far more numerous decisions about what to do are based on the situation at hand, and this is also the thing that players need to master to improve, learning which is best in a certain situation. This is also the reason you don't know which to build, the one that would be better depends on the game, so there is no clear answer that made you go "Aha! This is the correct one for this champion".
However, there are some in the first category answers that are more clear
Post-Script:
Cinderhulk on J4 and Vi would fall under the category of "You can, but why would you want to?" They do well at the clearing jungle, they don't need they help from the Cinderhulk for that, most who chose to jungle with them wanted to focus on ganking so the CDR and pen from warrior would help, and while they would frequently build some defense, the Cinderhulk's percent stat would be more efficient on a champion who would build a lot of defense.
: He clears well, he's always cleared well so trailblazer is a bit redundant IMO; it's not particularly needed.
On the other side of the coin though, Stalker's was made to give CC/ganking tool to those who didn't have one. Amumu has them already so it's not particularly needed. Skirmisher's is the dueling option, but Amumu doesn't "duel" so, (you might have guessed it) it's not particularly needed. It comes out odd because Amumu didn't need help in these ways; I would just go with Stalker's, but it's whatever.
: Shen is another case of "You can, but why would you want to?". You can get a Devourer on him and it would definitely help him, but if your goal in champion select was to pick someone to farm Devourer Stacks with for a half-hour, why pick Shen? I don't know.
= Devourer, Warrior, or Cinderhulk + Stalkers, Skirmishers, or Trailblazers?
= Cinderhulk + Stalkers, Skirmishers, or Trailblazers?
= Devourer, Warrior, or Cinderhulk + Stalkers or Trailblazers?
= Cinderhulk + Trailblazers. Stalkers, or Skirmishers?
= Devourer, Warrior, or Cinderhulk + Stalkers, Skirmishers, or Traiblazers?
= Cinderhulk or Devourer + Stalkers or Skirmishers?
