Can carbanion rearrange to attain more stability?
Sooner or later anionic rearrangement name reactions should turn up in your organic chemistry classes. I will present two in this post.

The first and arguably most well-known are the Wittig rearrangements; these can proceed as [1,2] or as [2,3] rearrangements.[1] The Kürti/Csakó states that:[1]


  The [1,2]-Wittig rearrangement proceeds via a radical-pair dissociation-recombination mechanism, while the [2,3]-Wittig rearrangement is a concerted, thermally allowed sigmatropic process proceeding via an envelop-like transition state in which the substituents are pseudo-equatorial.


This means that strictly speaking only the [2,3]-Wittig rearrangement could be considered an actual carbanion rearrangement as per your question but many chemists may also consider the [1,2]-Wittig rearrangement one since it effectively does generate a rearranged and thereby more stable anion. Both reactions are shown in scheme 1.


Scheme 1: the [1,2] (top) and [2,3] (bottom) Wittig rearrangements.

The second example has recently gained significant applications in synthetic organic chemistry; it is the same general reaction except it employs a silyl ether instead of an alkyl ether. Here, the stability of both sides of the reaction is similar so both the forward and backward direction are known; these are the Brook and retro-Brook rearrangements (see scheme 2).


Scheme 2: The retro-Brook rearrangement (top) and Brook rearrangement (bottom).[2,3]

West et al. who reported the reactions in scheme 2 note the interesting effect that the amount of butyllithium added influences the equilibrium. If substoichiometric or catalytic amounts are added, the normal Brook rearrangement takes place with the silyl ether being the more stable species. Under excess butyllithium conditions, the retro-Brook rearrangement takes place indicating that under those conditions the alcoholate is the more stable species.[3]

I mentioned that arguably the Brook rearrangement is more common in organic synthesis; this is due to Amos B. Smith’s anion relay chemistry or ARC.[4] Smith uses priviledged synthetic intermediates that can give alcoholates upon an initial reaction. The previously introduced silyl group migrates towards the alcoholate, liberating another carbanion which can subsequently participate in another nucleophilic attack. This allows multiple anion-dependent transformations to occur in a single reaction vessel.



References:

[1]: L. Kürti, B. Csakó, Strategic Applications of Named Reactions in Organic Synthesis, Elsevier Academic Press, 2005, p. 490.

[2]: L. Kürti, B. Csakó, Strategic Applications of Named Reactions in Organic Synthesis, Elsevier Academic Press, 2005, p. 64.

[3]: R. West, R. Lowe, H. F. Stewart, A. Wright, J. Am. Chem. Soc., 1971, 93, 282–283. DOI: 10.1021/ja00730a065.

[4]: A. B. Smith III, M. Xian, J. Am. Chem. Soc., 2006, 128, 66–67. DOI: 10.1021/ja057059w.