Regression Testing

I do.  However the focus of my work is a bit narrower than the scope of your question - I'm usually in a development cycle.

In the development cycle:

I start out with the release notes. 

Recompile all rules, looking for deprecation notices.

Then run through the set of functionality for the application being built - following the requirements traceability matrix.
Its pretty much the same as regression testing to a shift in design->implementation.

With an upgraded MST, same process. Release notes, then regression test against requirements.

In an existing application:
Migrate a copy of Prod back to Dev
Tie out between the two environments.

Upgrade Dev

Validate requirements against the matrix

Tie out data with Prod.

Success in repeating this process with QA strengthens the hypothesis that the upgrade is valid before proceeding to prod.

Each organization will formalize these outlined steps. 

I hope this helps.