The Royal Society for the Prevention of Accidents (RoSPA) collects statistics in the UK. I haven't read the documents in detail (you have to register to download...), but there is a summary page here: http://www.rospa.com/road-safety/advice/pedal-cyclists/facts-figures/
Mechanical failure doesn't seem to be common, it isn't even mentioned in the summary. Most injuries happen in a collision with a vehicle, so this is not the kind of crash you're asking about. The page says that, for adults, "16% of fatal or serious cyclist accidents reported to the police do not involve a collision with another vehicle, but are caused by the rider losing control of their bicycle."
Of course, "losing control" can mean a lot, including mechanical failures, and some of the collisions will also be caused by things like failing brakes.
As far as I know, there is a lack of data as bicycle crashes are normally not investigated in detail, unless there is a court case (but these are then usually collisions where guilt or liability has to be established).
From experience and general discussions it seems to me that the main mechanical failure that people have is in the brakes - the brake cable snapping, or worn brake pads (or the wrong kind of brake pads). The problem is that these issues may come without obvious warning - the brake may work well, but then suddenly fail completely when you need it. But you can reduce the risk a lot by good maintenance - check the brake pads and cables regularly for wear, and make sure both brakes are in good order so you always have one even if the other fails.
I've also heard stories of the front wheel coming off. However this is really rare and happened when they did a jump (e.g. jumping on a pavement, but they weren't going so fast). Not really on a road, as the wheel is pushed into the dropouts by gravity. The reason was really that the wheel nut wasn't tightened; so check that the wheel nuts or quick releases are fastened; again good maintenance reduces the risk. If you have a quick release and often leave your bike unattended, you may want to put a cable tie around the lever and the fork so that you can see if some joker fiddled with it.
If you have disk brakes, it is possible that the wheel is pushed out of the dropouts when braking, although this is more a design fault in some early designs and I think not so much of a problem any more.
Other types of mechanical failures (frame breaking, handle bar coming off etc) can happen in principle, but they don't usually happen so suddenly without warning, you'll usually notice the problem long before it's dangerous as the handling of the bike changes. For example, a crack in the frame may make the bike quite wobbly, but it doesn't suddenly disintegrate, and people have gone on for ages without crashing (or even noticing). I have once cycled for about hundred kilometers with several cracks in my rear rim.
So, mechanical failures can of course happen, but there are only a few types of failures that will be really dangerous (i.e. happen suddenly without warning), and much can be avoided by basic maintenance and generally checking your bike regularly.