Does MEMS accelerometer stability deteriorate over time?
The short answer is yes; they do change characteristics with time - and I have measured them to do so.  On advice from the manufacturer of the device I work with (in this case, VTI, now a division of Murata), most of the ageing occurs in the first year.

As far as what changes, the most obvious is bias.  I have seen drift from 3mg to 10mg on the SCA3100, though the maximum specification is 30mg over its lifetime (since the max bias is specified to be 70mg, and bias over temperature is specified to be 40mg).  These are 6-sigma specifications so few will ever reach that magnitude.

There is no obvious change to the amount of noise. Take this claim with a grain of salt - this statement is "by eye" and not properly based on data or statistical testing.

As far as other specifications is concerned (e.g. scale factor, cross-axis sensitivity), I gave no data, measurement or intuitive feel as to the magnitude of drift, if indeed there is any appreciable drift.

Other devices, of course, will have different characteristics, but I have no data to say what they may be.  I suspect the cheaper the accelerometer, the more it will age with time, but I have no evidence of that.

There are also short-term ("in-run") changes to the bias, even if the temperature is perfectly constant. If this is of interest to you, look for allan variance accelerometers on Google for a start on what drift characteristics to look for.