Your first clue to detecting a broken cymbal starts with the smallest variation with the cymbals normal sound. Check the cymbal for any sound difference, this could be a dulling of the crash, decrease in sustain or any different wacky sound.
What you want to look for are cracks. The first place to check for cracks would be on the surface of the cymbal (where you strike it with your stick). Small cracks may form and not be as visible as large cracks, especially if there is dirt or corrosion on the cymbal. Discolorations on the cymbal are ‘hot spots’ for cracks, they will form there quicker than on a cleaner surface. Another common place for crack would be around the bell, where the cymbal is mounted to the stand (also called key-holing).
Ride cymbals are harder to find cracks in because they are usually heavier/thicker than crashes/splashes. But the cymbal will start to show the same symptoms of one that is cracking.
Another tip for finding cracks is using touch. Simply run your fingers around the edge of the cymbal. You’re going to be feeling for any imperfections on the surface. This is a main area to find cracks because of the contact with your stick. You can do the same with the surface of the cymbal. Check the top and bottom. Sometimes you can feel a crack before you can see it.
In extreme cases, you’ll hear the crack. When you hit the cymbal, the vibration will cause the edges of the crack to rub against each other. When that happens, you’ll get a nasty buzzing/grinding noise added to your cymbal.