A 30,000 year old vase was found. What isotope would be best to confirm its age?
Carbon-14 works up to an age of 50,000 years with ordinary methods of sampling and measurement (https://en.wikipedia.org/wiki/Radiocarbon_dating).
The Wikipedia article goes into some detail about the method, including the effects of atmospheric variation in C-14 and other factors for which we must correct. Special sampling and measurement techniques can be used to extend the range to 75,000 years. Though limited, the range of C-14 or radiocarbon dating is wide enough to cover most matter that contains organic or carbonaceous material.