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["Lon J. Rombough" <lonrom@hevanet.com>]

What they usually mean by unstable is that the resulting mutant isn't all the "new" type.  It can be a mix of cells from the unmutated and mutated types.  This is usually known as a chimera.  For instance, a shoot can have normal cells in one layer and mutant cells in another and still look like the mutant type.  But an injury or other cause might make new shoots originate from the unmutated tissue and that type grows out.   Chimeras occur in natural mutants as well.  
There are ways to sort out plants that are all one type of tissue, but it usually isn't done until or unless the plant shows itself to be a chimera, and it can be a  rather tedious process.
You can make a plant genetically unstable this way by other processes besides mutation.  Tissue culture is one way.  
Simple heat shock will do it sometimes, too.  That is, exposing the plant to excessively high temperatures, usually over 110 F.  
Coleus normally has a square stem, but exposing shoots to high heat will make some of them produce hexagonal shoots.  Those aren't stable, though, and when you cut off the shoot, the new shoots revert to square stems.
Mutation breeding is an amazing topic to study.  Much of the work is on natural mutants, too.
-Lon Rombough
Grapes, writing, consulting, my book, The Grape Grower, at http://www.bunchgrapes.com  Winner of the Garden Writers Association "Best Talent in Writing" award for 2003. 
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On Aug 30, 2007, at 3:02 PM, Doug Woodard wrote:

would be cautious about assuming that all or most sports are caused by 
natural radiation.Cell division is complicated and Murphy's Law is 
pretty general.

Back in the 1970's, the Summerland BC station of Agriculture Canada 
publsihed a booklet on its fruit cultivars which mentioned that 
cultivers developed by neutron bombardment of (I think) seeds of older 
cultivers, had been found to be unstable. I think that Compact Lambert 
was one.

I have no idea whether mutations produced by other forms of radiation 
tend to instability; I suspect not.