>>20303150I have done extensive research for non-destructive at-home testing before jumping into precious metals. I think it's very important to have some knowledge in this regard and I'm surprised it's rarely talked about here.
That being said, I've never done the specific gravity test simply because I never had the need to and I'm also too lazy to try. Checking coins is also much easier than ingots, especially when blistered, and I 99% of what I have is in form of coins.
Anyway, measuring weight and dimensions of a coin (or an ingot) is the starting point. Once it passed those tests, whose values are easily found online, you should always do another test. The ping test is my favorite: it's quick and easy, and there are smartphone apps to help you with that ("Pingcoin" and "Precious coin tester"). I wouldn't just trust my experience with the sound, as it differs from coin to coin (see how short is the golden maple leaf sound compared to other 24K coins like the buffalo). Also you don't need that tool, as you can simply use your fingers to hold the coin or balancing on a tip like pic related and use a pencil or something to ping it; however I reckon it's very handy.
Also check the paramagnetic tests for silver and gold, where you can use a decent magnet with a precision scale or a plate filled with water. You need to buy a neodymium magnet, but it's easier to set up than a specific gravity test and it doesn't require math.