Which solar system objects have the least stable solar orbits? My research is not giving me anything definitive. Are the Apollo asteroids orbits less stable than the Centaurs or any Trojans, or is another class of objects in less stable solar orbits? On what timescale are those orbits unstable?
The least stable orbits are likely the temporarily captured orbiters of Earth and other planets. These are bodies that have been captured by the Earth/Moon gravity well and move from solar orbits to terrestrial orbits. They normally have complex orbits that take them well beyond the orbit of the Moon and most don't last long.
Most such objects are small, typically less than 1 m in diameter. There are probably about 1000 such bodies with sizes of 10cm and up orbiting the Earth. Normally, they are undetected but occasionally a larger object will be discovered.  Most recently, 2020 CD3 was captured in the winter of 2015-16 and escaped Earth's gravity in May 2020.
The orbits of such temporary orbiters are not stable, they normally escape the Earth/Moon gravitational well after a few years, and return to heliocentric orbits.