Pretty sure you could use both yeah?
it's definitely OR! Make Head or tail of something is to be able to understand it
I've only seen or heard it as plural: I couldn't make headS or tailS of it.
You should really use 'nor', because it's negative, i.e. "neither one nor the other", as opposed to "either one or the other".
it should be "nor" - because the sentence is negative.
Quite the opposite, it should be "Heads OR Tails" instead of NOR because "I couldn't make heads nor tails" would be a double negative.
Grammatically, it would be "head nor tail", as in "I could make neither head nor tail of it." However, the idiom is "head or tail", `and should not be manipulated based on context. So "I couldn't make head or tail of it" is correct.