The "Search" function was added on 2/24/2013. It is a feature that allows a user to send their Space Trio (the Galaxy Diver Lindwyrm, Starweaver Dragon, and Comet Breaker Amphithere) out to search for items or an egg. After 30 items have been collected any further items will be duplicates.
One can use this feature by visiting their dragon's Action page and choosing "Send on a search". The cooldown time varies, depending on how many sets of Space Trio dragons a user owns. Each set of the dragons reduces the cooldown time by 1 day until you reach the minimum wait of 2 days.
With this new feature came the release of the Supernova Dragon, a dragon only available through Search.
Dragon Search Badge
Most of the time, a user's dragons will bring back items to collect. These items can be viewed by clicking on the Treasure Chest icon inside of the user's Badge Book. There are 30 in total.
"Search" Dragon: The Supernova Dragon
The Supernova Dragon is an extremely rare dragon that can only be found by using the "Search" function. There is a very slim chance of a user's dragons to bring this egg back with them from a Search.