How many of the digits in a mean of 12.3456789012 are worth reporting?

A computer program tells me that a mean value is 12.3456789012, but how many of these digits are significant (the rest being random junk)? Should I report: 12.3?, 12.3456?, or even 10 (if only the first digit is significant)? There are several rules-of-thumb but, surprisingly (given that the problem is so common in science), none seem to be evidence-based.

Here I show how the significance of a digit in a particular decade of a mean depends on the standard error of the mean (SEM). I define an index, D_M that can be plotted in graphs. From these a simple evidence-based rule for the number of significant digits ('sigdigs') is distilled: the last sigdig in the mean is in the same decade as the first or second non-zero digit in the SEM. As example, for mean 34.63 ± SEM 25.62, with n = 17, the reported value should be 35 ± 26. Digits beyond these contain little or no useful information, and should not be reported lest they damage your credibility.