Which is the most stable measure of central tendency?

Stability means that there will be less variation between random samples drawn on the same population.

With categorical data, you may not have a choice, the mode is the only measure of central tendency that will be meaningful.

With measureable, numerical data, the mean may be the only meaningful measure of central tendency, even though the median may show less variation.

Some data may be assumed to have a skewed distribution, such as the price of homes, or incomes. The more stable and meaningful value for skewed distributions is the median, as a few high numbers can have a large impact on the estimate.

See related links. You can find more information on central tendency by doing a search on the internet.