Stability is predominantly a matter of how the wood has been cut (e.g. quartersawn versus plain/flatsawn). Tension, or reaction wood, can happen with any species as such is a product of either the conditions the tree was subject to in life (such as on a hill or mountainside, exposed to high winds, growth constrained in one direction by another tree, etc.) or can be created by faulty kiln drying (case hardening).
Wood is the product of a living plant and all wood is subject to movement when exposed to changes in humidity, though some cuts, such as plain/flatsawn, greatly exaggerate the movement by constraining movement in some dimensions while forcing movement in another, leading to cupping in accordance with grain and ring orientation. Woods with a very high resin content will tend to move more slowly in reaction to changes in humidity, but they still move and will continue to move until equilibrium is achieved with their ambient environment. When looking for woods that minimize humidity induced moisture dimensional changes, look for woods with a low shrinkage rate (typically under about 3%) from green to oven-dry (such tables are available from the
U.S. Forestry Products Laboratory, USFPL).
Some example woods with low shrinkage rates (2-3.4% radial) but by no means an exhaustive list:
(Common Name, Radial %, Tangential %)
Bigtooth Aspen, 3.3, 7.9
Butternut, 3.4, 6.4
American Chestnut, 3.4, 6.7
Horse Chestnut, 2.0, 3.0
Balsam Poplar Cottonwood, 3.0, 7.1
Silver Maple, 3.0, 7.2
Striped Maple, 3.2, 8.6
Black Willow, 3.3, 8.7
Cedars (most species below 4% radially), (2.2-4.6), (4.7-6.9)
Balsam Fir, 2.9, 6.9
Grand Fir, 3.4, 7.5
Subalpine Fir, 2.6, 7.4
White Fir, 3.3, 7.0
Balsa, 3.0, 7.6
Butternut, 2.4, 6.4
Eastern Hemlock, 3.0, 6.8
Eastern White Pine, 2.1, 6.1
Sugar Pine, 2.9, 5.6
Redwood (young/old growth), (2.2/2.6), (4.4/7.4)
Primavera, 3.1, 5.2
Afrormosia, 3.0, 6.4
Balsa, 3.0, 7.6
African Mahogony, 2.5, 4.5
Cocobolo, 3.0, 4.0
Honduran Mahogany, 3.0, 4.1
Purpleheart, 3.2, 6.1
Brazilian Rosewood, 2.9, 4.6
Indian Rosewood, 2.7, 5.8
Iroko, 2.8, 3.8
Jelutong, 2.0, 4.0
Teak, 2.5, 5.8
Complicating choices a bit is that while there is a correlation between radial and tangential shrinkage rates the correlation is not perfect and so some woods will shrink/expand considerably more on one axis versus another, which may influence one's choice depending upon how the wood will be used in its final application. In some applications you may wish to go with the lowest tangential shrinkage rate instead. There are a relative few woods with unusually low combined radial and shrinkage rates but they tend to be either exotic woods (such as Mahogany, Rosewood, Cocobolo, Iroko) or domestic woods less commonly used in fine woodworking (such as Cedar and Redwood).
But, hopefully, the above give you some ideas.
. But If you don't want it to move make your guitar out of a man made material like plastic or ... 