Which of these are correct?
I have seen all except the last being used.
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One could argue that they are all correct in modern usage. "Hi" is often substituted for "high" in certain (video game) contexts; similarly, "Lo" for "low". This usage is so pervasive these days as to be acceptable now IMHO.
If you wanted to be a pedant, I'd plump for "high-score", or even "highest score".
I have no problem with Hi-score or High-score, although I don't see the need for a hyphen.
As for Highscore, I don't see the need to eliminate the space and mash it together as one word (unless it's a variable name or something, in which case the rules of standard English hardly apply anyway).
I would avoid Hiscore, however. At first glance, it's too easy to erroneously parse as his core instead of hi score.
Finally, if you did break it into two words, you might want to use title case (i.e., High Score), depending on the context.