So how is all this nuclear waste currently dealt with?  There are many ways, but a lot depends on the nature of the waste and state/ federal regulations.  Some of it is stored in cooling ponds, others are turned to a glass waste form, it's put into large containers and buried, and that just lists a few ways.  In nearly all of those cases, the waste is treated as whole, without regard to what's actually in the waste itself.  However, there is a big need to separate the high-level radioactive waste (like cesium) from the low-level radioactive waste (like rare earth elements) so that waste can be stored more efficiently, or even recycled.

An important class of minerals that have been explored for waste storage are zeolites and related minerals.  zeolites are like natural molecular sponges; they are porous on the molecular scale (microporous), and can absorb elements from a liquid into the solid crystal.  When they do, they also have to release the element that already resides in it.  Think of a dish sponge that has been colored green by food dye.  If you want to wash off the green dye, you have to replace it with fresh water.