Most woods will benefit from stabilizing to a degree. It's not always about making soft, punky, otherwise unusable wood hard and durable.
Much of the wood I sell is for use by knifemakers as handle material and they overwhelmingly prefer their wood and other materials to be stabilized. Stabilized woods are not affected by temperature and moisture the way unstabilized woods are. They don't shrink or swell depending on a climate's humidity or other climatic conditions.
Stabilized woods tend to polish easier and show off the figure better.
They are much harder and less susceptible to denting, cracking or checking.
They are much more water resistant and are much less susceptible to oil or chemical stains.
As for what woods are good candidates for stabilizing? Anything that is not extremely dense or oily. Also wood with a moisture content above 10% should not be stabilized.....6% is ideal.
Some of the woods that I regularly stabilize: All Maples, Walnuts, Amboyna Burl, Afzelia Burl, Curly Koa, Buckeye Burl, York Gum Burl, Red Mallee, Brown Mallee, Corrugata Burl, Concinna Burl, Spalted Hackberry and others.
If you visit my website you'll see a more detailed list of about 35 species (give or take).
Some species to avoid: Ironwood, African Blackwood, cocobolo, Thuya Burl, Camphor Burl, Eucalyptus Burl, Olive wood, and Snakewood.
Anyway, this is far from a complete list but I hope it helps.
Feel free to contact me if I can answer any specific questions.
Best Regards,