In Russian, if I want to say something like "just in case they will be useful to you," is it correct to say "на всякий случай вам пригодятся"? - да, правильно ( It's correct) Would "буде" or "авось" be better than "на всякий случай"? - it's slang, you could use last one but it depends on situation

"just in case they will be useful to you," - "На всякий случай, вдруг вам понадобится" or just "На всякий случай" "авось" is something different. It's like expression of hope that something will happen. "Авось пронесёт!" - "I hope it will be OK!"

На всякий случай. Вдруг вам это пригодится. Это может пригодиться. Ещё мы говорим:"На всякий пожарный (случай)". )))

You can say "just in case they will be useful to you" more interesting - Пригодится на "чёрный день"! - it means - useful in extreme cases!!! "авось" is not a word, it is a Russian philosophy :-))) it is like "it will never happen, but who knows!!!"

Currently "Буде" word is not used to communicate in Russian! Perhaps the word remains in the small villages, but osnovnovnaya part of the population does not use that word!