When the diode "avalanches," which simply means that it starts pulling current due to excess voltage, it may or may not break depending on whether the current is sufficiently limited by the surrounding circuitry and whether the diode is "avalanche rated."
Vendors will generally note in their data sheet whether a device is avalanche rated. Avalanche rated devices will act like zener diodes and will only die if too much heat accumulates at some point on the die. Then, localized melting will typically poison the junction and the diode will usually be shorted. That is, unless sufficient current is available. Then, the device and package will typically burst apart as the current burns it open.
Some devices, like Schottky rectifiers die due to punch through. This is somewhat like an ESD failure in appearance. It occurs due to a sudden breakdown of the devices ability to hold off voltage in one place. When this happens, you can see a physical pit in the metallization at the point of failure. To offset the occurrence of this problem, manufacturers will build in a zener device in the structure. This is typically about the outer edge and serves to avalanche at a lower voltage, thus protecting from punch through.
If your really interested in rectifiers, their idiosyncrasies, and the inside on manufacturers, I can give you more than you'd probably care to hear :), so feel free to message.