This is not something I feel is really common however I have known people to watch certain domain names to wait until they expire and then jump on them and purchase them before the owner of it can renew it themselves if they wish to.
Have you ever or would you ever do something like this? Has this ever happened to you and if it did how did you handle it?
Yes absolutely. There is a whole industry around this.
Companies that offer to catch domains before they are released again to the public are called drop catchers. Snapnames and Namejet both allow you to do this, as well as many domain registrars (like Godaddy's backorder service).
There is also a weird period of time when a domain may have expired at Godaddy but can still be renewed, and it is nevertheless placed for auction. If the domain expires completely, the auction winner gets the domain, if not the domain is cancelled.
You can get some really good domains at good prices if you get in when no one is looking
I have never done this myself as I never seem to be quick enough however I know people that have and they say that it works really well for them in terms of them either getting a domain they really want or in terms of getting a good domain that has aged quite well to make some extra cash.