How do you determine isotope stability?
With difficulty. You measure the number of nuclear disintegrations per second and then calculate the half-life of the substance.
The definitive method of testing for the presence of radioactivity is to use suitable procedures to make measurements with a suitable instrument. However, the operative word is suitable.
Nuclei decay by emitting one or more neutrons, protons, alpha particles, beta particles, X-rays, or γ rays. Some of these forms of radiation are extremely difficult to detect, and different instruments are sensitive to different types of radiation. There is no such thing as a "universal" instrument that will work in all circumstances.
Only 90 isotopes are expected to be perfectly stable, and an additional 163 are energetically unstable, but have never been observed to decay. Thus, 253 isotopes are stable by definition.
The rate at which a radioactive isotope decays is measured by its half-life — the time it takes for one-half of the nuclei of a radioactive material to disintegrate. Half-lives for various radioisotopes range from 10⁻²⁴ s (hydrogen-7) to 10²⁴ years (tellurium-128).
A stable nucleus, then, has a half-life greater than 10²⁴ years.