  Perturbations of giant magnons and single spikes in a $2+1$ dimensional
$\mathbb R \times S^2$ background spacetime are analysed. Using the form of the
giant magnon solution in the Jevicki-Jin gauge,the well-known Jacobi equation
for small normal deformations of an embedded time-like surface are written
down. Surprisingly, this equation reduces to a simple wave equation in a
Minkowski background. The finiteness of perturbations and the ensuing stability
of such giant magnons under small deformations are then discussed. It turns out
that only the zero mode has finite deformations and is stable. Thereafter, we
move on to explore the single spike solution in the Jevicki-Jin gauge. We
obtain and solve the perturbation equation numerically and address stability
issues.
