It might be useful then to define what "Stable" means for our purposes of discussion then. The stability of a material with regards to woodwind making is, for me, defined by the following factors:
1) Stability Lengthwise: Does it tend to warp from top to bottom, the way we often see boxwood do?
2) Stability around the bore Eccentrically: Will the piece warp or "Oval" in that way that makes the tenons not fit into the sockets properly?
3) Stability around the bore Concentrically: How much will the wood shrink/contract around the bore axis in the way that causes all the headjoint cracks on flutes with metal linings?
4) Ability to resist stress: How much shrinkage concentrically can the flute resist before cracking? Plus, how well can the wood tolerate differentials in moisture content - which is to say, if you pick up a flute that is relatively dry and play it for an extended period, will it crack?
5) Stability of the grain of the wood Vertically: How much does the grain "raise" over time as the instrument is played causing the previously smooth bore to become rough which impacts tone and playability? Also does it absorb so much moisture the tenons swell an inordinate amount causing ill fitting joints?
To the woodwind maker these should all be considered important measures of stability as they all impact the how the instrument functions over time. Every wood has various proportions of the qualities listed above, and all are inherently unstable it's a matter of picking your poison so to speak.
Some Examples:
Boxwood: On the downside, it can be very warpy, particularly when not properly seasoned and stabilized properly. It also tends to swell an awful lot when played for long periods. Again this can be largely mitigated if the proper precautions are taken during the design and making of an instrument from the wood. On the positive, Boxwood tends to maintain a very smooth bore if the instrument left the shop that way. And it is very much more resistant to cracking than Grenadilla, Cocus, most of the rosewoods, and many other woods.
Grenadilla and Cocus ( lumping them together for our purposes because they are very similar): These don't swell or warp much (cocus I think swells a bit more in my limited, pre allergy experience), the grain of the bore stays very smooth if well finished initially, but over time this wood is highly likely to crack if used over any sort of metal liner. Some may say "Not so!" for Grenadilla, but I have seen lots of cracked heads and barrels by current makers, much of this no doubt due to poor seasoning practices, and a few makers are a bit notorious for their poorly seasoned wood, but still, we'll see plenty of cracked Grenadilla headjoints over the years - hear me now and believe me later
(SNL reference for those old enough to remember Hans and Franz) Also, Grenadilla is nowhere near as resistant to cracking when played dry as boxwood is. I know of more than a few instances where someone cracked a perfectly good grenadilla instrument by playing it for an extended period after the instrument had been sitting for months or years unplayed. I have yet to hear of this happening with a boxwood instrument, although I am not saying it can't happen.
Mopane: Seems pretty stable in general but (in my limited experience) it swells a bit during play and the grain in the bore tends to raise quite a bit impacting the sound.
Dogwood: Similar to mopane in that it stays straight, but it swells quite a bit, at least as much as Boxwood making for rather problematic fit of socket and tenon joints. Also the grain of the bore does raise some over time, not so much as Mopane but still, I find the swelling enough to impact tone unless the bore is re-reamed or sanded smooth.
So for me, Grenadilla is a fabulous wood for woodwind making, but it has as much instability overall as any of the other woods. Grenadilla's instability is just heavily weighted in the negative with regards to shrinkage concentrically and inability to handle the stress of radical moisture content differentials.
Anyway, that's where I'm coming from on the subject of stability and tonewoods. I was lucky enough to work at a shop that had used many different woods over many years and we got to see an awful lot of those instruments for regular services of cleaning, recorking, crack repair, thumb bushing replacements or adds, etc., so I got to see how those instruments held up over time. We kept a detailed record of every instrument made, so I was able to look up every instrument by serial number if I wanted to check the exact age, pretty cool, not to mention informative! In addition the owner of the shop had an extensive collection of antique woodwinds (many I never even saw) and basically what I came to understand from all of this is that over long periods of time, all instruments are going to have some problems regardless of the natural material used. Poor design and/or execution tends to contribute to more problems, but in the end I didn't see grenadilla instruments faring any better than boxwood instruments. YMMV
Loren