First instance imaged of an asteroid breakup. I have done asteroid confirmations at my club's site and do have an IAU observatory code for that site because of it. There are lots of images of comets breaking up due to the volatility of those objects. Comet Shoemaker-Levy 9 is one example of that when it broke up when captured by Jupiter's gravity and then the pieces slammed into the Jovian upper atmosphere providing a view of multiple "black eyes" visible for weeks with the small backyard telescopes. However, most of the asteroids are "rubble piles" of various sizes of rocks and dust. It is theorized that these rubble piles do break up and then re-coalesce due to the lack of ice found in them. Some asteroids do breakup due to the YORP effect or by getting too close to a large body such as the Earth or other planets.
Most of the early flybys of asteroids were of the cratered dust covered large rocks with some having a tiny moon around them. P/2013 R3, the asteroid pictured breaking up is an example of a rubble pile is designated as a comet. The designation given is a date coded type. The P before the slash stands for "periodical" usually reserved for comets. Since this asteroid is now acting like a comet, it gets a P. 2013, the year of discovery; R meaning Sept 1-15; 3, the order within that half-month of the discovery.
25143 Itokawa, is an example of a rubble pile asteroid. The number is the final designation of an asteroid or dwarf planet after much research has been done of that particular body and the name is the accepted name by the naming committee of the IAU. It was named after a Japanese rocket scientist. It was visited by the Japanese probe Hayabusa and the probe returned the first and only soil sample return of an asteroid to Earth. Below is the image of Itokawa taken by the Hayabusa space probe, showing a textbook image of a rubble pile. The rotation is along the length causing the peanut or dog bone shape of these objects.
Joelene