Examples where theory suggests something is possible but it never gets used in practice:

It is pretty easy to find examples of things that are solved in theory but either (1) are too inefficient to use in practice or (2) no one cares about. Examples: (1) (generic) zero-knowledge proofs, (2) undeniable signatures. In fact, look at any crypto conference and at least half of the papers will probably fall into one of these categories.

Examples where theory suggests that something is safe that is not safe in practice:

This question is a little vague, so I'm not sure if this answers it -- but there are plenty of examples of 'provably secure' schemes that get broken in practice because the security definition did not match the deployment scenario. In the last few years alone there were attacks on (provable variants of) SSH and IPSec, among others.

Examples of something in widespread practical use has little theory behind it:

I assume you mean in the crypto world, not in the general security world. A good example is DSS signatures, which have no proof of security.