If you want to secure your tweets against any issues that might happen to Twitter and its servers, consider backing up your Twitter shares. If you use Twitter as a bookmarking tool sharing the most interesting links you come across, you may want to create a searchable archive to be able to quickly find a link you remember sharing a few days or months ago.
Twitter search could be a good solution (there is also a handy from: operator that allows you to find all tweets from any user) but it is only able to work for recently shared links. Twitter search only stores tweets for up to 10 days. If you want to be able to search through older tweets, consider using one (or all) of these tools to create a searchable archive of your tweets or any other Twitter-powered databases.