For which fractions is it standard practice to specify using 'over' instead of '-ths'?

Solution 1:

There no sharp switch. I suggest, from experience compatible with the comments, that the transition is essentially complete by x/101, with some use below this and some exceptions.

Commonly-used fractions below x/100 almost always use the form similar to the ordinal numbers (e.g. inverse powers of two: x/64 would be x 64ths). I suspect that they are common because they're used in measurements, such as distances in inches. Uncommon denominators may use either as they approach 100.

I propose two reasons that work together: it starts getting harder to be clear, and some combinations don't work. "Two hundred and seventy-fifths" presumably means 2/175 but relies on hearing the s clearly after the fth. It could also be parsed as x/275 for unspecified plural x. A clumsy partial solution is "two one-hundred and seventy-fifths". Denominators ending with a 1 (except 11) don't help - even one twenty-first is less intuitive than surrounding numbers.

The biggest exceptions are major powers of ten, such as thousandths and millionths, followed by many of the hundredths ("one five-hundredth"). I've certainly heard one ten-thousandth used (I'd probably use one part in ten thousand, rather than "... over...", but that may say something about the context in which I use such fraction). These again are common, but for comparing magnitudes.

From experience, the major binary powers (after 64) don't work like the powers of ten. I supervise an experiment in which the smallest measurable signal is 3.3V/1024, and I know if I refer to "thousand-and-twenty-fourths" I'll only confuse people.

So you can't write a rule, you can only really try to capture what people use in practice, and this won't be universal.