If it's leaking, fix it. The end.
Ok, I'm kidding. Today I'm going to share some home inspection tips and tricks that homeowners can use to identify plumbing problems. You'll want to use a good flashlight for your plumbing inspection, as a lot of this work involves looking underneath sinks and tub drains.
Bathroom Sinks
Most homeowners already know about the more obvious leaks under bathroom sinks, but to really test your sink for leaks, fill the sink with water and then let it drain all at once. This test will force a large slug of water through the drain, and will often identify leaks that wouldn't otherwise be seen. Carefully watch the drain while performing this test. One of the most common leak locations at bathroom sinks is at the drain stopper; fixing this leak is usually as simple as tightening the nut.
If the bathroom sink drains slowly, it's usually the result of hair in the drain. Fix this by pulling the hair out with a Zip-It tool. What's a Zip-It? I'm glad you asked. It's an inexpensive, effective, and easy-to-use drain cleaning product invented by a Minnesotan. Click the following link to find out, but prepare yourself to see some absolutely disgusting photos of hairballs removed from drains: http://zipitclean.com/
Toilets
Stand at the toilet with the front of the bowl between your legs, and give the toilet a little nudge with your shin to make sure it doesn't rock or swivel. A loose toilet can lead to a leaking toilet.
Flush the toilet several times times and check behind, around, and under the toilet (if possible) for any leaks.
If you have a toilet that clogs frequently, replace it. I recommend using Consumer Reports to help decide on a toilet. Their team tirelessly tests toilets in the most tasteful manner possible to figure out which ones have the best flushing ability. I've trusted them in the past, and they haven't let me down. I'll leave it at that.
Showers/Tubs
I wrote a whole blog post about identifying shower leaks over five years ago, and not much has changed since: http://www.structuretech1.com/2009/07/finding-shower-leaks/ .