One of the most common problems is the barbell crashing in the clean (and even in the snatch to a lesser extent). There are two very simple ways to help correct this:
1. Meet the bar wherever it is
2. Hold your grip longer
First, you need to be wherever you've lifted the bar to meet it. People sometimes become infatuated with the idea of getting down quickly into the rock bottom squat. The ability to do so is great, but not if the bar is higher than that level. That is, you need to relocate your body directly under the bar, at whatever level you've lifted it to, not just "down" indiscriminately. I make the analogy to people sometimes that it's akin to either doing a normal front squat or overhead squat, or getting down into the bottom of the squat position and having someone drop the barbell onto their shoulders or into their hands - which is going to be more secure, stable and easier to recover from?
Second, you need to maintain your grip on the bar long enough. This is much more natural in the snatch, particularly if you're a lifter who holds the hook grip overhead. However, you should in any case be punching the bar up into the overhead position with a relatively loose grip, so even if you maintain your hook, you need to transition from a tight pulling grip to a loose pushing grip at some point. That point is ideally as the hand is turned over under the bar.
Likewise, with the clean, where the problem of prematurely releasing the grip is more obvious, the grip needs to be maintained until the elbows are moving up in front of the bar - really as long as possible without preventing a quick and complete fixing of the elbows/shoulders in the proper rack position. If you're flexible enough and built the right way, you can maintain a full grip or close to it even in the rack position--this is fine as long as your turnover is quick and timed properly. In any case, try to maintain your grip until the bar is on your shoulders.
These two things will improve your lifts considerably once you implement them correctly