What can go wrong with a GLIBC upgrade?

It is often possible to compile up a fresher libc, and have one piece of software use a different library to everyone else.

The biggest painpoint between libc versions is thread local storage, IIRC. If you're passing this milestone, all bets are off.

Having said all that - I would strongly recommend that you simply upgrade the distro to something modern. Doing piecemeal upgrades is likely to lead to the sort of pain that has you chasing your tail for months.

You can refer to the compatibility table to determine binary-compatible versions.

Usually it's not a problem to upgrade glibc to a newer version, but you should build it with the same configuration flags as used in your distribution (see spec file in the appropriate SRPM package).