What percent of crashes are due to bicycle failure?
There are some statistics available from Great Britain for Contributory factors for reported road accidents. Below are the 2014 figures for Pedal Cycles, the first column is number and the second is percent.



Note that only reported accidents where a police officer attended the scene are included in the statistics. This Wikipedia article has some details of the methodology used to collect the statistics.

The complete data set is available here. The data above is taken from the table RAS50005 - Vehicles in reported accidents by contributory factor and vehicle type, Great Britain, 2014.