How can I coerce some pops and crackles out of my base/stock 2012 XK?

 

Get the Mina exhaust system and you will have subtle pops and crackles when down shifting or "feathering" the accelerator That is what I have on my 12 XK vert. You will also get a deep throaty sound under heavy acceleration.

 

Trade it in for an F-type R!

 

Suggest the cheapest mod is to replace the central resonator with a shallow X pipe ala "F". Summit Racing sells them:
http://www.summitracing.com/parts/wlk-53716

The main muffler on the XK has an internal spring-loaded flap that opens up the exhaust at a certain gas level. My guess is that this is about 4,000 rpm or so with full throttle. I suppose one could cut the muffler open and replace the exhaust spring with something a with a bit lighter extension, but this gets pretty technical as spring longevity could become an issue.

You could also install a used (hopefully cheap) XKR muffler and modify its external valve operation.

Another active exhaust possibility is Dynomax VT mufflers. These are "active" mufflers with an internal spring-loaded valve. Their action is much more aggressive than the stock XK muffler. Also, they are cheap.