Over two million pets and other live animals are transported by air every year in the United States. Since 2005 all U.S. airlines that operate scheduled passenger flights have to file monthly incident reports. The U.S. Department of Transportation requires these reports concerning incidents involving pets that injured, were lost or died during transport.
In 2014 U.S airlines have reported the followning incidents:
We can conclude that pets travel safe and the chance of a serious problem for your pet during flight is really small. Your chances of winning the big prize in a state lottery are a little bit higher.
Source: Air Travel Consumer Report/Airline Animal Incident Reports, Issued: February 2015/U.S. Department of Transportation