  We derive general conditions for the existence of stable scaling solutions
for the evolution of noncanonical quintessence, with a Lagrangian of the form
$\mathcal{L}(X,\phi)=X^{\alpha}-V(\phi)$, for power-law and exponential
potentials when the expansion is dominated by a background barotropic fluid.
Our results suggest that in most cases, noncanonical quintessence with such
potentials does not yield interesting models for the observed dark energy. When
the scaling solution is not an attractor, there is a wide range of model
parameters for which the evolution asymptotically resembles a zero-potential
solution with equation of state parameter $w = 1/(2\alpha -1)$, and oscillatory
solutions are also possible for positive power-law potentials; we derive the
conditions on the model parameters which produce both types of behavior. We
investigate thawing noncanonical models with a nearly-flat potential and derive
approximate expressions for the evolution of $w(a)$. These forms for $w(a)$
differ in a characteristic way from the corresponding expressions for canonical
quintessence.
