When diagnosing audio problems, you need to keep in mind that each SID (C64 audio chip) sounds different. Their filter strength varies greatly even across the same revision and can make a huge difference in the audio output. The problem is that the musicians made the tunes for THEIR SID, so they may sound wrong or outright bad with another chip. In some cases so bad that you might mistake it for a malfunction of the SID chip. Another thing is that most newer tunes were made for the new SID model, 8580, so they may not sound right with the original model, 6581.
Both SID chips in the example videos are operating normally. The first chip has very light filters, while the second one has very heavy filters.
Although Trailblazer (Example 1) sounds pretty awful with the heavy filters, the SID chip itself is working fine. Its filters are simply "incompatible" with Trailblazer's tune. Compare it with the first chip's output and you'll notice a big difference.
Digital Delight 3 (Example 2) is another example where the heavy filter SID fails to deliver. In this case the filters seem almost completely missing. This is not an actual fault; the tune was simply made for a light filter SID, resulting in the heavy filters being almost inaudible.
Another oddness you may notice is the scraping "noise" that some SID chips - especially those with light filters - produce when playing back certain tunes. This is clearly audible in Example 3 - Bump Set Spike. The second SID plays it much more smoothly. Again this does not mean the first SID is faulty, it's just a normal side effect of its "sharp" filters.
Images & Audio (c) SOASC
The best SID test I have found so far can be found here:
It's a simple test program that can be directly typed in as BASIC code or downloaded from the site. The test plays a few different sounds, and completes in less than a minute. Listen to what your C64 sounds like, and compare it to the correctly played recordings available on the above site. If they sound similar, there is a good chance your SID is OK.
Stone Oakvalley's Authentic SID Collection has almost every SID tune available for download as MP3, and they were recorded from a real C64 (3 different SID revisions in fact). If the above test catches no problems but you still feel a certain tune or tunes sound wrong, download the tunes from SOASC and compare them to your C64's output.
If your C64 refuses to boot and you suspect a bad SID, you can try booting the computer without this chip. A C64 can operate just fine without the SID, there just won't be any audio. The chip is marked as U18 on the board, and is usually socketed, making removal easier.
If you have a newer model (C64C) that produces no audio output at all, check the fuse. If the fuse in a C64C goes bad, the machine will work normally but without sound. The older "breadbin" type C64s will seem completely dead if the fuse goes - the red LED will still light up, but not much else.
Another problem you may encounter with a C64C is the digital audio sounds bad or is very low volume. This is a known issue with the C64C short boards. You can do a custom fix for this by connecting a 330k Ohm resistor between pin 26 (Ext-IN) and pin 14 (ground) of the SID chip.