We have multiple servers with PG versions in 8.4, 9.2, 9.3, 9.4 and 9.5, that we want to upgrade to 9.6. We want to be sure that all applications will run smoothly after upgrade.
The only way I know to be sure is to is to set up a test instance of 9.6 and test your applications.
I would as matter of course also review the release notes for each major release since 8.4 to look for further gotchas.