Sanity Test on Oracle Upgrade Database
Unless your development or testing team already has a set of regression test cases, you are out of luck.  The possible things to test are astronomical.  Now, as others have pointed out, there are some things to look out for, like more restrictive privileges, and reviewing the "What's New" for both 11g and 12c.

One big example from the past that burnt a lot of people (including me) was upgrading from 9i to 10g; where developers relied on a "group by" clause in a select would always return the results sorted by the group by columns in 9i, but not always in 10g.  Not a bug; the SQL standard for group by says nothing about ordering of results; so a lot of code broke.  But again, only way to flush out those types of things are good regression test cases.

Now, after all that, I have not seen or heard about any 10g to 12c surprises, but there are a lot of wonderful new features that should be used when moving to  12c.