Units for vibration measurements

I am conducting research to determine vibration limits for cultural heritage objects (e.g. art, small sculpture, ceramics). The vibration source can be, for example, transport, construction in or next to a museum, or rock concerts in or next to a museum. As a former materials scientist and engineer in the aircraft industry working on component lifing, I am fully aware that vibrations are a form of cyclic load, and that such damage is cumulative. The limits have to be based on some form of an S-N (Wöhler) diagram for fatigue life. S is normally given in stress amplitude, but one could also use displacement/strain, which can be converted to stress. However, in the vibration monitoring world, vibrations are given either in velocity or acceleration, and sometimes power density spectra.

For buildings, vibration criteria are generally given in units of velocity, but in transportation, they are often given in g, and sometimes PDS.

I have been using velocity as a measure for S in the S-N diagrams, since that is what building engineers use, and it is the closest I can get to deformation (one derivative). If I give a vibration limit in velocity (and duration) and the frequency range in which it is valid, then I am indirectly giving a deformation, which is what one sees in building codes.

In the transport industry, acceleration (g) appears to be used more often. I assume that this has to do with the fact that the cause of damage is more likely to be shock than continuous vibrations.

So my question: which unit, velocity or acceleration, correlates better to damage due to vibrations, and why?