Let's say I made a contest to eat bananas in one day. I ate 12 bananas today. That would be:
My bananas score
Or
My banana score
Let's say I made a contest to eat bananas in one day. I ate 12 bananas today. That would be:
My bananas score
Or
My banana score
Obviously you'd have to use this in a context where people understood the game, and that you were speaking about the game.
I would use:
My banana score.
I see this as being similar to saying something like:
My banana count.
I.e. It's your "banana count" for the day.
This would be similar to the term "head count", which is used when counting the number of people in some context. In as much as you wouldn't say "heads count", I don't think you would say "bananas count", and therefore you wouldn't say "bananas score".
Further to this, unless you had a scoring system more complex than just a direct count, you may just use the word "count" rather than the word "score" - it's actually quite idiomatic in scenarios where people invent these kind of simple contests to have with each other, e.g., many people compare their Fitbit "step count".
There could be another situation, where you might say:
My Bananas score.
This would be the case if you had named the game Bananas.
The correct rendering would be
My banana score.
There is no such thing as a "banana score", but that isn't to say you can't create the premise, and then use it. 11 years ago there was no such thing as an "Uber rating", and now there is.
So, assuming you have explained the premise - that you are eating bananas, keeping a count of how many you eat, and "scoring" yourself based on how many you consume in a day - you can refer to this score.
I think you should use the singular "banana score" for two reasons:
This is consistent with other similar scoring systems. As I mentioned, your Uber rating is based on your use of Uber taxi cabs. Even though you may have ridden in may Ubers, it isn't your "Ubers rating" - it is singular. Also consider the Motion Picture Association of America film rating (it isn't "films"!), and Five Star Hotels (not "stars"!)
"Bananas" can also mean "crazy", so calling it "my bananas score" could mean your score is crazy, or ridiculous.
It is possible in English to create compound nouns in which one noun effectively acts as an adjective limiting another noun. For example, we say "information technology" or "medical technology" to distinguish between technologies used for different purposes or using different aspects of knowledge.
This is a grammatical feature of English. If for example I say "My golf score yesterday was awful," it will be understood because almost everyone in the English-speaking world knows there is a competetive game called golf in which each player earns a score. But the compound word "banana score" will be meaningless because virtually no one has heard of your games of eating bananas (or eating partners, which is likely illegal in many jurisdictions).
A compound noun is usually an abbreviated way of saying something that avoids the use of prepositional or participial phrases. Like many abbreviations, it is meaningful only if the audience can supply the missing information.
People will understand if you say
My score in my fraternity's banana-eating contest was 18
because you have supplied contextual information. Unless they were participants or onlookers, they will believe you to be talking nonsense if you say
My banana score was 18.
The issue of whether "score" should be singular or plural depends on on the number of contests that you are referring to rather than the numerical value of the score itself.