So what is the procedure for checking a bike? I've been very into bikes my whole life, never had one fail, and I confess I do not know how to test anything other than looking and riding one.
>Brakes: Look to see that your brake pads are not worn
>Chain and Cranks: Pull on your cranks to see that they are not loose and look to see that the chain is not rusted and it is free of gunk
Newer bikes add:
>Quick Release: Make sure all quick releases are closed
and it's a good idea to:
Check: Take a slow brief ride to check that your bike is working properly
and of course the recommendation has always been to take a bike in to a bike shop in the spring before resuming riding.
The notable mechanical failure I had was a broken gear on the rear cassette (and this in a lifetime of riding which has included a couple of bike tours, a lot of general riding, and a couple of years of almost exclusively commuting by bike (which I'm hoping to get back to)) --- fortunately, was able to pick up the pieces, set the bike on a particular gear, then continue riding --- getting hit by a car on the first day of summer vacation when I was 14 was far more momentous.
I don't have any specific knowledge about this, but one of the things I do when buying bicycles second hand is lift them up and give them a good shake. Any noise should be easy to attribute and once addressed, it should be quiet. This helps discover weak joints between components.
I also visually inspect the frame – especially on the underside and around the bottom bracket and head tube, and look for any deformations or cracks.
I have always wanted to try to apply bending forces to the frame and listen for creaking, but I haven't figured out a good technique to do so with a stationary bicycle. The best I can achieve is a test ride where I try things like accelerating hard on a high gear, doing little front-wheel hops, etc.