why is it important to find stability of supergravity solutions? I want to know about the importance of the stability,
generally,why is stability important in super-gravity?
and why do we like these solutions?
 A: (expand my comment as an answer)
First, in supergravity theories (e.g the Kaluza-Klein type) the vacua solutions may be infinite or many.
This immediately poses a problem for a physical theory which needs some uniqueness properties.
Then the property of stable vacua (aka stable solutions) is used as a condition for selecting among the possible soluions for the ground state of the theory. (for example a reference on Kaluza-Klein supergravity, section 5.2)
When a vacuum state is not stable it is refrerred as metastable (as in thermodynamics) or false vacuum. This is a local minimum but not the lowest energy state, even though it may remain stable for some time. As such it is meta-stable and can revert to the lowest energy state at some time.
From wikipedia article on stable vacua and relation to cosmology:

For decades, scientific models of our universe have included the
  possibility that it exists as a long-lived, but not completely stable,
  sector of space, which could potentially at some time be destroyed
  upon 'toppling' into a more stable vacuum state. The
  Standard Model of particle physics opens the possibility of
  calculating from the masses of the Higgs boson and the top quark,
  whether the universe's present electroweak vacuum state is likely to
  be stable or merely long-lived.
...This catastrophic bubble of "true vacuum" (per quantum models)
  could theoretically occur at any time or place in the universe, which
  means (because the bubble of "true vacuum" will expand at the speed of
  light) the end of such a false vacuum could occur at any time.

