It's the handbrake.
Nope, really, it is!
At the back where the squeak is coming from (usually the right) there is a rubber grommet with the handbrake cable, spray some lithium grease or similar on it, and that will sort it.
Mine has been fine for at least 3 months since I did it.
A bit of searching will reveal some pictures if you have issues following my crap description.
You do not need to jack the car up, but it will be easier for the first time to see where it is. Don't need to remove the wheel though.
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MX5 1.8 Sport
Might have the wrong forum here...